Heat oil in a nonstick skillet over medium heat. Add mushrooms, onion, and garlic; saute 7 minutes. Sprinkle flour over vegetable mixture; stir well. Stir in milk; cook over medium heat, stirring constantly, until thickened and bubbly. Stir in broccoli, salt, and pepper. Let cool. Stir in egg yolks and cheese.
Beat egg whites and cream of tartar at high speed of an electric mixer until stiff peaks form. Fold beaten egg white mixture into broccoli mixture.
Coat a 10-inch springform pan with cooking spray; place 1 sheet of phyllo in pan (keep remaining phyllo covered). Coat phyllo with cooking spray. Layer 9 more sheets on top of first sheet, coating each with cooking spray and fanning each slightly to the right. (The overhanging sheets will form a circle around the pan.) Gently press phyllo into pan, forming a shell; fill with broccoli mixture. Top with remaining 3 sheets, coating each with cooking spray and fanning each to the right. Fold edges over top to enclose filling. Coat top of phyllo with cooking spray.
Bake, uncovered, at 400° for 25 minutes. Cover and bake 35 minutes; let stand 10 minutes. Serve warm.