Heat butter in a skillet over medium heat; cook and stir the onion and mushrooms with salt and pepper until the mushrooms are dark and juicy, about 5 minutes.
Remove skillet from heat and stir in the bread crumbs. Set aside.
Place each chicken breast half between two sheets of heavy plastic (resealable freezer bags work well) on a solid, level surface. Firmly pound the chicken with the smooth side of a meat mallet to an thickness of about 1/4 inch.
Lay a flattened chicken breast out onto a work surface, and place a slice of ham on the chicken. Spoon about 3 tablespoons of the mushroom stuffing onto the chicken breast, and top with about 2 1/2 tablespoons of the Havarti cheese.
Roll the chicken breast around the stuffing, wrap 2 slices of bacon firmly around each roll, and secure with toothpicks.
Place the rolls into a 9x13-inch baking dish.
Bake in the preheated oven until the chicken is no longer pink and the stuffing is melted and hot, about 35 minutes. An instant-read thermometer inserted into the center of a roll should read at least 165 degrees F (74 degrees C).
Remove the chicken rolls from the oven, and place an oven rack about 6 inches from the oven's broiler heat source. Turn on the broiler, and broil the rolls until the bacon is brown, 5 to 10 more minutes.
