So I just bought my first house and in lieu of changing loads of outlets in the kitchen for GFCI outlets I went with two 20A GFCI breakers. Upon going to install the breakers I found that the cable which feeds to my kitchen contains 2 hot conductors and 1 grounded ("neutral") conductor.
Is there any way I can connect both GFCI breakers to this cable? I tried splitting the neutral between both breakers and found that they both immediately trip.
You need to install a 20 ampere double pole GFCI breaker, instead of two single pole breakers.
There isn't a way that you can use a GFCI breaker on this circuit.
The way a GFCI works, roughly speaking, is by measuring the current leaving on the hot wire and comparing it to the current returning on the neutral. The two currents should be identical – any difference is indicative of current "escaping" via a different ground path, possibly through a person.
The circuit you describe uses two out-of-phase (if they are wired correctly) hot wires that share a neutral. This configuration saves a bit on the cost of the wire, since you can run two circuits on three conductors instead of four, but it means that you loose the ability to use a GFCI on the circuit since the return current on the neutral will almost certainly not match the current on the hot wire(s).
Since you've been working on this circuit it would be a good idea to confirm that the two hot wires (the red and black ones) are indeed wired out of phase. They should be on adjacent poles in the breaker panel, or if they aren't for some reason make sure that they are on breakers that are out of phase (you can check this either by inspection or by measuring the voltage between the breakers, it should be around 240 Volts).
The safety concern with this type of circuit is that you potentially have two loads that, if the both failed, could expose you to 240 Volts between them.
