1. Preheat oven to 425°.
2. To prepare crust, weigh or lightly spoon flour into a dry measuring cup and spoons; level with a knife. Combine flour and next 3 ingredients (through baking powder) in a food processor; pulse 2 times or until combined. Combine 1/4 cup oil and 3 tablespoons ice water in a small bowl. With processor on, slowly add oil mixture through food chute, and process until dough is crumbly. Sprinkle dough into a 9-inch pie plate coated with cooking spray. Quickly press dough into an even layer in bottom and up sides of pie plate. Place crust into preheating oven, and bake for 10 minutes.
3. To prepare filling, place squash in food processor (do not clean from dough), and process for 1 minute or until squash is finely chopped. Heat a large nonstick skillet over medium-high heat. Add 1 tablespoon oil to pan; swirl to coat. Add squash and onion to pan; sauté for 9 minutes, stirring occasionally.
4. While squash cooks, combine half of cheese (about 1/3 cup), eggs, 1/4 teaspoon salt, and 1/4 teaspoon pepper in a large bowl; stir in squash mixture. Remove crust from oven; spoon squash mixture over crust, and spread evenly. Return tart to 425° oven; bake 9 minutes.
5. Return pan to medium-high heat. Add remaining 1 tablespoon oil to pan; swirl to coat. Add pancetta; cook 1 minute or until beginning to brown. Add mushrooms; cook for 7 minutes or until browned. Stir in remaining 1/4 teaspoon salt and 1/4 teaspoon pepper. Add wine; cook 1 minute or until liquid almost evaporates. Remove tart from oven. Arrange the mushroom mixture evenly over top of tart; sprinkle with the remaining 1/3 cup cheese. Return tart to 425° oven. Bake 3 to 5 minutes or until cheese melts.