Begin by preheating your oven to 350 degrees F. Spray a 9x13-inch baking dish with cooking spray. Fill with the bread, ham, peppers, and cheese. Mix to evenly combine.
In a large bowl, whisk together the eggs, milk, Dijon mustard, garlic powder, fresh thyme, salt, and pepper. Pour this mixture on top of the bread mixture. Gently press down the bread with a spoon. Sprinkle with shredded Parmesan cheese.
Bake 50-55 minutes until the eggs are cooked through. Cover the pan with aluminum foil if the top begins to brown too quickly. (If preparing the casserole in advance, cover the baking dish with aluminum foil and refrigerate overnight. Bake as directed.)