Preheat oven to 425°. Unroll pie crust onto a lightly floured surface, roll to a 12-in. circle. Transfer to a 9-in. deep-dish pie plate; trim and flute edge. Refrigerate while preparing filling.
In a large skillet, heat butter over medium-high heat; saute onion, mushrooms, broccoli and peppers until mushrooms are lightly browned, 6-8 minutes. Cool slightly.
Whisk together eggs, cream, salt and pepper. Sprinkle 1/2 cup cheese over crust; top with spinach and vegetable mixture. Sprinkle with remaining cheese. Pour in egg mixture.
Bake quiche on a lower oven rack 15 minutes. Reduce oven setting to 350°; bake until a knife inserted in the center comes out clean, 25-30 minutes. (Cover edge loosely with foil if necessary to prevent overbrowning.) Let stand 10 minutes before cutting.
Pastry for single-crust pie (9 inches): Combine 1-1/4 cups all-purpose flour and 1/4 tsp. salt; cut in 1/2 cup cold butter until crumbly. Gradually add 3-5 Tbsp. ice water, tossing with a fork until dough holds together when pressed. Wrap in plastic and refrigerate 1 hour. Editor's Note Based on your reviews we decided to take a second look at this recipe. After retesting we decreased the volume of egg mixture and the volume of veggies to fit in the dish better, increased the salt and added pepper for more flavor and started baking it at a higher temperature on a lower oven rack to get the bottom of the crust cooked through.