Over medium heat in a medium saucepan, sauté the onions and carrots in 2 Tbsp. butter until softened, about 8-10 minutes. Add the beer and simmer until almost all of the liquid has evaporated, about 30 minutes. Remove from the pan and set aside to cool.
Preheat the oven to 350 degrees.
Whisk together the eggs, milk, cream, salt and pepper.
Roll out your pie crust in between sheets of plastic wrap for easy rolling to the size of your pie shell. Remove one side of the plastic wrap and invert onto your pie shell. If your shell is too warm and tears apart, pop the crust in the pie shell inside the freezer for 10 minutes to firm it back up. The plastic wrap will remove easily without tearing this way.
Layer the carrot and onion mixture on the bottom of the shell first, then add the cheese, then the egg mixture. Top with sliced ham.
Bake in the oven for about 45 minutes or until the crust is golden brown and the quiche is set in the center. Let cool for 10 minutes before slicing.