How can I get my layer cake to cook more evenly/not burned?
In a conventional oven, the majority of the heat is at the top.

You don't mention lowering the oven rack to compensate for the change in height. I'm fairly certain that would solve your problem.

Center rack is normally recommended - if you used a higher rack then that's definitely the reason it burned. But even if you used the center rack, you might need to lower it for a particularly tall cake. You might also need to extend the cooking time slightly to compensate for the lower effective temperature.

Note that this doesn't apply to convection ovens, which have a relatively constant temperature throughout.