Heat a large skillet over medium-high heat. Add butter and when it foams, add shallots, celery, carrots and garlic, and season with salt and pepper. Cook 3 minutes then add stock and stir 2-3 minutes more, reducing liquid to 1/4 cup. Add hot sauce and chicken, adjusting heat level to taste. Cook to heat through and until liquid is almost absorbed, 1-2 minutes more.
Heat a griddle or large, cast-iron skillet over medium heat. Butter or spread mayo on 1 side of all of the bread slices. With the greased sides facing out, build sandwiches: bread, cheddar cheese, chicken filling, blue cheese crumbles, American cheese and bread.