Heat butter in a frypan 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 frypan 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 5mm.
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 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 baking dish.
Bake in the preheated oven until the chicken is no longer pink and the stuffing is melted and hot, about 35 minutes. Remove the chicken rolls from the oven and place an oven rack about 15cm from the grill. Grill the rolls until the bacon is brown, 5 to 10 more minutes.