Preheat oven to 375 degrees F (190 degrees C). Grease a 9x12-inch baking dish.
Place a chicken breast between two sheets of heavy plastic (resealable freezer bags work well) on a solid, level surface. Firmly pound the chicken breast with the smooth side of a meat mallet to a thickness of 1/4 inch. Repeat with the other chicken breasts.
Melt the butter in a small skillet, and cook the mushrooms until they begin to brown, about 8 minutes. Set aside. Beat egg in a shallow bowl. Place the cracker crumbs in another shallow bowl.
In a bowl, mix together the mushrooms, cream cheese, 1 cup of Cheddar cheese, and Parmesan cheese; spread the mixture generously over the chicken. Roll the chicken breasts up, dip each into beaten egg, and roll the bundles in cracker crumbs. Place into the prepared baking dish. Sprinkle the bundles with the remaining 1/4 cup of Cheddar cheese.
Bake in the preheated oven until the chicken is cooked through, the filling has melted, and the crumbs are golden brown, about 30 minutes. An instant-read thermometer inserted into the center of a roll should read at least 165 degrees F (74 degrees C).
