Heat oven to 375°F. Spray 1 1/2-quart casserole with cooking spray. Place rolls on cutting board; let stand 10 to 15 minutes to thaw. Meanwhile, in medium bowl, beat eggs, milk, basil, salt and pepper until well blended.
Cut rolls into 3/4-inch cubes; place in bowl with egg mixture. Stir to coat. Stir in broccoli, ham and cheese. Pour into casserole.
Bake 40 to 45 minutes or until puffed, golden brown and knife inserted in center comes out clean. Let stand 5 minutes before serving.