Don't go overboard with charge rate
Charge rate is programmable - You don't have to charge at 60A. And probably can't without a service upgrade. But we can avoid that cost.
Here's a great video on a very common error made by novice EVers that has cost them a ton of money and probably even burned a few houses down. Please watch a few minutes of it, starting at 28:15.
If you can get to 32:55, all the better. That delivers perfectly usable charging at sensible cost, that will probably fit on your panel.
Feel free to watch the whole video if you like. The first part drones on a bit about watts and volts and stuff, but it grows the beard.
Your house probably can't support 60A charge rate
Another common blunder by novice EVers is to slap a giant charging circuit on an already-loaded service. This bets the farm that the main breaker will do its job - that's not how electrical is done! For a large EVSE you are required to do a NEC Article 220 Load Calculation for all your house's loads. (though a great many, present company excepted of course, ignore that.)
That will reveal how much spare capacity exists for level 2 charging. If it's not enough, you'll either need to a) get a service upgrade, b) get special EVSEs capable of monitoring other loads in the house and reducing EV charge rate to prevent overload, or c) scale back your charge rate ambitions to what the service can support.
Generally, installing an EV circuit requires pulling a permit. The city will want to see your Load Calculation and will require that your installation stay within it. Power companies often want you to report EVSE installations as well.
It's not that bad
I know that many novice EVers are gripped by fear that their charging won't be enough. I understand. But it's a false fear. Think about it, what are your lease terms? Most leases give 12,000 miles a year. Well assuming you only drive workdays, that's 48 miles a day. You could do that with the 120V level 1 charger lol. But, we want to have a significant margin for unusual periods of use, so we want something like 140-170 miles in 14 hours. Well, that is exactly the 20A/240V charging discussed at 32:55 in the video. And you are much, much more likely to fit that on a service like yours than a 60A behemoth.
And even if you managed to have two 250 mile days back to back, 5-15 minutes at a Supercharger in the evening on the way home the first night can close the gap. The Supercharger doesn't have to fill the car to full, just get it far enough along for the home unit to finish.
The Tesla Wall Connector can be configured for any breaker size: this is done when commissioning the EVSE. I would advise setting it for a 20A breaker if you're not willing to do the number-crunch and figure out what your circuit can handle
I really want 60A, though, and I'm willing to pay for it.
I would suggest holding off until you get more experience with EVs and discover how unnecessary it is. But if you insist, there are three ways to go about that.
First, you can crunch your NEC Article 220 Load Calculation and see if it actually comes out to less than 140A. Good luck lol.
Second, you can pay for a service upgrade to 300A or 400A.
Third, you can use a load-sensing EVSE that monitors other circuits and automatically throttles back EV charging when those other circuits are in use. This is becoming a feature of "smart panels" like the SPAN - however we expect this technology to fall greatly in price. But more typically you will have an EVSE which adds a current sensor to your panel. It clamps other wires to see what their load is, and makes sure those loads plus the EVSE don't exceed a certain amount.
For instance, let's suppose your 60A EVSE would fit on your Load Calculation if the water heater and dryer were deleted. You have the EVSE's sensor clamp the water heater and dryer wires and tell the EVSE "Don't let all of you together exceed 60A". You'd be able to sit in the car and watch this; when the 4.5kW water heater kicks on, charge rate would drop from 11 kW to 6.5 kW.
However, most EVSEs which use this technology expect to clamp the main service wires. That won't work on your panel because of how it's laid out.
A similar and simpler thing can be done simply by identifying a 30A load with a socket (this means, your dryer) and extending that circuit into the garage with another socket. This will require cord-and-plug connection to the EVSE.
Cord and plug vs hardwired
Novices often think they need a cord-and-plug. No, Tesla Wall Connectors can be hardwired, and at 60A must be hardwired. Additionally, many states have adopted NEC 2020. Unless they have deleted this rule, any 240V socket in a garage is require to have GFCI protection, necessitating a 240V GFCI breaker ($150-ish). So it is better to avoid the socket if feasible.
If you share the dryer circuit, the socket is required, and upgrading the dryer circuit to "grounded" (if it's 3-prong) is also required, and so the GFCI is required too.
As far as wires, each wire has a maximum size. You can always use a larger wire than is required. Tesla's Wall Connector allows only copper connections but 75C thermal limit, allowing wires other than NM or UF higher amps.
- 12 AWG copper allows 20 amp breaker.
- 10 AWG copper has a maximum 30 amp breaker.
- 8 AWG NM and UF (Romex) max out at 40 amps.
- 8 AWG other types of copper cable, 50A.
- 6 AWG NM and UF (romex) max out at 55A. Not adequate for 60A.
- 6 AWG other types of copper, 65A.
- 4 AWG NM and UF (Romex) max out at 70A.
Space for an EV circuit
You have a Murray panel. The only correct breakers for that panel are Murray MP or Siemens QP (which is simply a rename of MP). Alien breakers can damage the panel. Now I gather the A/C contractor put that Cutler Hammer BR250 breaker in there? That should be changed to a Siemens Q250.
Further, I note that both of the 30A breakers have red-black wires - no white wire on the breaker, so clearly these circuits have neutral. Since they do, they need a common trip breaker (one side overloads, both trip). That Eaton BQ was not one, by the way.
Now looking at the panel label it appears every breaker is permitted to have tandem breakers. You certainly could compress the two 30A/240V breakers into a Siemens Q23030CT2, a quad with common trip both inner and outer.
While you're installing quadplex breakers, I call your attention to all the single breakers with red wires on them. These are called Multi-Wire Branch Circuits and they are wired dangerously wrong. MWBCs must be phased so there is 240V across the two hot wires in the same cable! Further, there MUST be handle-ties across the two handle throws. So you will need to read up on MWBCs, follow the hot wires to identify your MWBCs, and install quadplex breakers appropriately. However, if the MWBCs don't have 240V loads on them, they don't need common-trip breakers. Depending on what Siemens offers, it may also be possible to do it with handle-ties.