Cut broccoli into spears, removing leaves and tough ends. Blanch in boiling water then cool quickly in ice water. Chop into approximately ¾-inch pieces and set aside.
Melt butter in a large skillet over medium heat. Add ground red pepper and sauté onion, garlic and red bell pepper.
In a bowl, beat the eggs and combine with mustard and black pepper then mix in milk.
Place half of the bread cubes in a well buttered 13×9-inch baking dish. Add layers of half the amounts of the onion mixture, broccoli, ham and cheese. Add second layer of bread cubes and pour half of the egg mixture over the layers. Repeat layers with remaining onion mixture, broccoli, ham and cheese. Pour the remaining egg mixture over the assembled layers.
Cover and refrigerate at least 6 hours or overnight.
Bake uncovered at 350°F for 50 to 60 minutes or until the top is golden brown and bubbly.