Terminology first: Your well was drilled, not dug. No one in Vermont would dig a domestic water well deeper than about 30 feet. You probably have a steel well casing about 6 inches in diameter sticking out of the ground, right? That’s a drilled well.
The odor is from hydrogen sulfide, a gas that is produced by anaerobic sulfur bacteria which metabolize sulfates and iron in your water and release the gas as a waste product. The bacteria themselves are totally harmless to humans. These bacteria generally do not exist at great depths until and unless they are introduced by human activity such as well-drilling or debris falling into a well. Once introduced and allowed to thrive unchecked, these bacteria can eventually spread over months or years for great distances through the water bearing strata to other properties.
You did not initially have rotten-egg water because there were no sulfur bacteria in the aquifer until your well was drilled and the bacteria were introduced to that depth. They took some time to multiply and now you can smell their waste product.
Next issue: did the well driller install a sanitary well cap? This is supposed to let air in and out of the well pipe but prevent bugs and vermin from getting into the well. It should be bolted to the top of the well casing, not just resting on top. Failure to install a sanitary cap is a violation of well drilling regulations in some states.
Question: Did the well driller sanitize the well before leaving the job site? Every state has its own standards. I don’t know Vermont’s regulations, but I’m your neighbor in New Hampshire and it’s a requirement here. The mechanics of drilling a well always introduces bacteria into the water-bearing stratum, or aquifer. It’s unavoidable. The well driller, when they finish the job and just before they cap the well, is supposed to add a measured amount of bleach or another sanitizer, circulate the water for a time, then pump out the well until there is no detectable bleach. A licensed well driller who fails to perform this important step is risking his license because it is in the State’s best interest to prevent bacterial contamination of water bearing strata.
If you can’t get your well driller to remediate this situation, you should consider reporting them to the licensing agency in your state.
You can remediate the sulfur bacteria yourself using bleach. You’ll need to know the diameter of the well pipe, the depth of the well and the depth of the water surface in the well. That will determine how much bleach to use. It will probably be a few gallons. Please look up instructions for sanitizing a well from a reputable source such as a state agency (any state) or a university. Don’t follow instructions from some random guy on the internet like me. (!)
WARNING WARNING: Do not use scented bleach, or no-splash bleach (which contains detergent) or anything but regular bleach.
The instructions will have you pour bleach down the well, then you’ll connect a hose from your water system’s spigot, open the valve and direct the pump’s flow back down the well. You’ll run the water that way for hours. Don’t worry about your pump burning out; continuous running does not harm a well pump.
The instructions may have you run bleach-y water through your plumbing also, before you shut off the hose’s valve and leave everything to sit for a certain recommended time. This time is the chlorine’s “contact time” which is needed to be sure all the beasties are dead. During this time, some of the bleach-y water at the bottom of the well will diffuse into the aquifer. Hopefully, it will kill any sulfur bacteria that have spread into the aquifer from the well.
Finally, the instructions will have you open the hose’s valve again, but direct the water onto the ground, not back into the well. The well should have its sanitary cap reinstalled at this time to keep it free of contaminants. After several hours of pumping, you can check the water and see if the chlorine odor has dissipated. If not, keep pumping until the water is chlorine-free. You could use a chlorine test kit if you wish, but the nose knows.
Remember to flush out your plumbing too. You may get some black water if there are any pipes involved. Just flush till it’s clear and odor free.
You’ll probably have delicious water again, at least for a time. If the sulfur odor comes back, you didn’t kill all the beasties, or they have spread so far that they can’t be killed.
If that happens, your water can be treated to remove the hydrogen sulfide. This involves a non-pressure tank to hold several months’ worth of a weak bleach solution, a metering pump to inject a very small amount of bleach solution into the water stream as you use it, a large pressurized contact tank where the water will spend some time in contact with the chlorine so that the hydrogen sulfide can be oxidized, and then a backflushing activated carbon tank to absorb any excess chlorine. This is a proven system that is low-maintenance and works very well. It will give you good-tasting water again.
But before you use a carbon filter for any purpose, you must test your water for radon. Carbon filters in high-radon water quickly become radioactive and must be treated as hazardous waste. The good news is that radon-in-water removal, if needed, is highly effective. My water has 80 times the state limit before treatment but no detectable radon after treatment. Don’t panic if you have radon.
I hope you have a better understanding of your stinky water now!
EDIT: Ecnerwal has provided a link to Vermont's web page for hydrogen sulfide in well water.
Vermont well sanitizing instructions. (Includes a chlorine calculator.)