How to Make It
Place oven racks in top third and lower third of oven. Place a large baking stone on lower rack; leave in oven throughout recipe preparation. Preheat oven to 400°F. Drizzle cut side of garlic head with 1/2 tablespoon oil; wrap tightly in aluminum foil. Bake garlic in preheated oven on top rack until garlic cloves are softened, about 45 minutes. Unwrap foil, and set garlic aside.
Toss together eggplant, squash, zucchini, 5 tablespoons oil, 4 teaspoons salt, and 2 teaspoons black pepper in a large bowl. Divide mixture evenly between 2 rimmed baking sheets, and spread into a single layer. Place baking sheets on top and bottom oven racks. Bake at 400°F until vegetables are just tender, 15 to 20 minutes, rotating baking sheets after 10 minutes. Set aside to let cool 15 minutes.
Heat remaining 2 tablespoons oil in a large Dutch over medium. Add onion, bell pepper, remaining 2 teaspoons salt, and remaining 3/4 teaspoon black pepper; cook, stirring occasionally, until onion mixture is softened, about 10 minutes. Squeeze roasted garlic head to release cloves. Add garlic cloves and tomatoes to onion mixture. Cook, stirring occasionally, until mixture has thickened slightly and tomato juices have mostly evaporated, about 12 minutes. Remove from heat. Stir in basil, thyme, crushed red pepper, and roasted vegetables. Season to taste with salt and black pepper, and let cool completely, about 1 hour.
Unwrap 1 chilled pie dough square, and place on a lightly floured work surface; roll into a 24- x 16-inch rectangle (about 1/8 inch thick). Fit dough into an 18- x 13-inch rimmed baking sheet (or half sheet pan), allowing excess dough to hang over edges. Trim dough overhang to 1 inch; discard scraps. Chill, uncovered, at least 15 minutes or up to 1 hour.
Unwrap remaining chilled dough square, and roll into a 24- x 16-inch rectangle. Cut lengthwise into 1-inch-wide strips. Spread vegetable filling in an even layer in chilled pie shell in baking sheet. Brush edges of dough lightly with some of the beaten egg. Arrange dough strips in a lattice pattern on top of filling, leaving about 1/2 inch between strips. Trim dough strip overhang to 1 inch, using some of the trimmed longer strips to complete lattice pattern as needed. Fold bottom dough overhang and lattice strip overhang under, and gently press together on pan edges. Freeze pie, uncovered, 20 minutes. Reduce oven temperature to 375°F.
Brush dough lightly with remaining beaten egg. Bake pie in on bottom oven rack on top of preheated baking stone until crust is golden brown and filling is bubbly, 1 hour to 1 hour and 15 minutes. Transfer pie to a wire rack. Let cool at least 30 minutes before slicing. Serve warm or at room temperature.