Place 3/4 cup warm water in the bowl of an electric stand mixer fitted with the dough hook attachment. Add flours; mix on medium speed until combined, about 20 seconds. Cover, and let mixture stand 20 minutes.
In a small bowl, mix yeast and remaining 1/4 cup water; let stand until bubbly, about 5 minutes. Add yeast mixture, 1 tablespoon oil, and 1/2 teaspoon salt to flour mixture; mix until a soft dough forms. Place dough in a large bowl coated with cooking spray; place plastic wrap coated with cooking spray directly on surface of dough. Chill 24 hours.
Let dough stand 1 hour, covered, until dough reaches room temperature. Place dough on a lightly floured surface; roll into a 14- x 11-inch rectangle. Press dough into bottom and partially up sides of a 13- x 9-inch metal baking pan coated with cooking spray.
Place a baking sheet in oven on bottom rack. Preheat oven to 450°F. (Do not remove baking sheet while oven preheats.)
Toss together eggplant, zucchini, squash, 1 1/2 teaspoons oil, and remaining 1/4 teaspoon salt on a separate rimmed baking sheet; spread in a single layer. Bake at 450°F for 15 minutes or until tender, turning once halfway through.
Heat 1 1/2 teaspoons oil in a large nonstick skillet over medium-high. Add onion and bell pepper. Cook, stirring often, until tender, about 4 to 5 minutes.
Spread pizza sauce over dough, leaving a 1/2-inch border; top evenly with spinach. Sprinkle mozzarella over spinach. Arrange eggplant, zucchini, and squash slices alternately over cheese. Top evenly with onion and bell pepper mixture. Brush remaining 2 tablespoons oil over dough border. Place baking pan on preheated baking sheet; move to middle oven rack. Bake at 450°F for 22 to 23 minutes. Cut into 6 squares.