Place bread, half the garlic, 1/2 cup of shredded Asiago cheese, and melted butter in a food processor. Pulse until bread is turned into crumbs. Set aside.
Bring a large pot of water to a boil. Add 1 tablespoon of salt and the ziti. Cook according to package directions. Drain. Set aside.
Preheat oven to 375 degrees and grease a 9x13-inch baking dish.
To the same pot that you cooked the pasta in, add the 5 tablespoons of butter. Add onion and cook for 5 minutes. Add the remaining garlic and red pepper flakes and cook 1 minute.
Stir in the flour and cook 1 minute.Whisk in the wine.
Gradually whisk in the milk and the chicken broth. Bring to a boil.
Add chicken and simmer for 5 minutes. Remove from heat and stir in remaining 2 cups of shredded Asiago cheese and black pepper. Check for seasoning and add salt to taste.
Place broccoli in a microwave-safe bowl with 1 tablespoon of water. Cover with plastic wrap and microwave for 2 to 3 minutes.
Add ziti and broccoli to the pot with the sauce and stir to mix. Transfer mixture to prepared baking dish. Scatter the bread crumb mixture on top.
Bake for 20 to 25 minutes. If bread crumbs start getting too browned, cover loosely with foil.