Weigh or lightly spoon baking mix into dry measuring cups; level with a knife. Combine baking mix and oil in a medium bowl. Cut in butter with a pastry blender or 2 knives until mixture resembles small peas. Add ice water, 1 tablespoon at a time, tossing with a fork until moist. Place between 2 sheets of plastic wrap; roll into an 11-inch circle. Place in a 9-inch deep-dish pie plate. Fold edges under and crimp; chill.
Heat a large nonstick skillet over medium-high heat. Coat pan with cooking spray. Add sausage to pan; cook 6 minutes or until browned, stirring frequently. Drain on paper towels.
Coat pan with cooking spray. Add bell peppers and onion; cook 3 minutes, stirring frequently. Stir in squash, thyme, and crushed red pepper. Coat vegetables with cooking spray; cook 4 minutes or until squash is almost tender, stirring frequently. Stir in 2 tablespoons water; cook 1 minute, scraping pan to loosen browned bits. Remove from heat; stir in sausage.
Combine egg, egg whites, and milk in a medium bowl, stirring well with a whisk. Spoon sausage mixture into prepared crust. Pour egg mixture over sausage mixture.
Bake at 375° for 35 minutes or until set. Remove from oven; sprinkle with cheese. Cut into 8 wedges.