To prepare crust, weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine flour, 3/4 teaspoon pepper, and 1/2 teaspoon salt in a large bowl; cut in butter and shortening with a pastry blender or 2 knives until mixture resembles coarse meal. Gradually add ice water, 1 tablespoon at a time; toss with a fork until flour mixture is just moist. Divide the dough into 3 equal portions. Roll one portion into a ball (about 5 ounces). Combine remaining 2 portions; roll into a larger ball (about 9 ounces). Gently press each portion into a 5-inch circle on plastic wrap. Cover with additional plastic wrap. Chill at least 30 minutes.
Preheat oven to 425°.
Remove greens from beets; discard stems. Chop greens; set aside. Wrap beets and potatoes in foil. Bake potatoes at 425° for 1 hour or until tender. Bake beets at 425° for 1 hour and 20 minutes or until tender. Cool completely. Reduce oven temperature to 400°. Peel and slice beets and potatoes into 1/4-inch-thick rounds.
Heat a large Dutch oven over medium-high heat. Coat pan with cooking spray. Add onion; sauté 4 minutes. Add garlic; sauté 1 minute. Add beet greens, spinach, 1/4 teaspoon salt, and 1/4 teaspoon pepper; sauté 8 minutes or until liquid evaporates. Place spinach mixture in a large bowl. Cool completely.
Combine eggs and egg white in a medium bowl, stirring with a whisk. Reserve 2 tablespoons egg mixture. Add remaining 1/2 teaspoon salt, remaining 1/4 teaspoon pepper, and cheeses to bowl, stirring well.
Unwrap and place larger portion of dough on a lightly floured surface. Roll dough into a 14-inch circle. Press dough into bottom and 2 inches up sides of a 9-inch springform pan coated with cooking spray. Arrange half of beets and half of potatoes over dough. Top with half of spinach mixture and half of cheese mixture. Repeat layers with remaining beets, potatoes, spinach, and cheese mixture.
Unwrap and place smaller portion of dough on a lightly floured surface. Roll dough into a 10-inch circle. Place dough over filling; press edges together. Cut several slits in top of dough to allow steam to escape. Brush top of dough with reserved 2 tablespoons egg mixture. Bake at 400° for 40 minutes or until crust is browned. Let stand 10 minutes.