Pulse bread, 2 minced garlic cloves, 1/2 cup Asiago, and melted butter in food processor until coarsely ground. Set aside.
Adjust oven rack to middle position and heat oven to 400 degrees. Bring 4 quarts water to boil in large pot. Add 1 tablespoon salt and ziti to boiling water and cook until nearly al dente. Drain in colander and rinse under cold water until cool.
Melt remaining butter in empty pot over medium heat. Cook onion until softened, about 5 minutes. Add remaining garlic and pepper flakes and cook until fragrant, about 30 seconds. Stir in flour and cook until golden, about 1 minute. Slowly whisk in wine and cook until liquid is almost evaporated, about 1 minute. Slowly whisk in milk and broth and bring to boil. Add chicken and simmer until no longer pink, about 5 minutes. Off heat, stir in remaining cheese until melted.
Microwave broccoli, covered, in large bowl until bright green and nearly tender, 2 to 4 minutes. Stir cooked broccoli and drained ziti into pot and season with salt and pepper. Transfer to 13- by 9-inch baking dish. Sprinkle with bread crumb mixture and bake until sauce is bubbling around edges and topping is golden brown, 20 to 25 minutes. Cool 5 minutes. Serve. (Casserole can be assembled, minus bread crumbs, 24 hours in advance. Bring to room temperature before adding bread crumbs and baking.