Preheat the oven to 375 degrees F (190 degrees C).

Melt butter in a skillet over medium heat. Add onion and mushrooms; season with salt and black pepper. Cook and stir until mushrooms are dark and juicy, about 5 minutes. Off heat, stir in bread crumbs. Set aside.

Place chicken breasts, one at a time, between 2 sheets of heavy plastic (resealable freezer bags work well) on a solid, level surface. Firmly pound breasts with the smooth side of a meat mallet to a 1/4-inch thickness.

Lay 1 breast onto a work surface; place 1 slice ham on top. Spoon about 3 tablespoons mushroom mixture onto ham; top with about 2 1/2 tablespoons Havarti cheese. Roll breast to enclose filling. Firmly wrap roll with 2 strips bacon; secure with toothpicks. Place roll in a 9x13-inch baking dish. Repeat with remaining breasts, ham, mushroom mixture, Havarti cheese, and bacon.

Bake in the preheated oven until chicken is no longer pink and fillings melted and hot, about 35 minutes. An instant-read thermometer inserted into centers of rolls should read at least 165 degrees F (74 degrees C).

Remove rolls from the oven. Carefully set an oven rack about 6 inches from the heat source. Turn on the broiler and return rolls to the oven. Broil until bacon is brown, 5 to 10 minutes.