1. Put yeast in a small bowl; add water and sugar. Gently whisk to combine. Let stand until foamy, about 5 minutes.
2. Mix together flours and salt in a large bowl. Form a well in center of flour. Pour oil into the well and, with clean hands, rub oil into flour. Add yeast mixture to flour; mix to combine. Knead dough on a lightly floured work surface until smooth and elastic, about 6 minutes. Form dough into a ball, place in an oiled bowl, cover with plastic wrap and a clean dish towel, and let dough rise at room temperature for 90 minutes.
3. About 30 minutes before dough is ready, preheat oven to 500° with rack in center and pizza stone on rack. Once the oven reaches temperature, heat stone for at least 30 minutes.
1. Heat 1/2 tablespoon oil in a large skillet over medium-high heat. Add mushrooms; cook until tender, about 2 minutes. Transfer to a small colander set over a bowl to drain excess liquid. Add remaining oil to skillet, and add spinach; cook until wilted and deep green, about 2 minutes. Transfer to a plate. Place tomatoes in a bowl and, with clean hands, break up into small pieces. Season sauce with salt.
2. When dough is ready, gently remove from bowl and cut into 6 pieces. Work with 1 piece of dough at a time, keeping unused dough covered with plastic wrap. Using both hands, roll into a ball, then press into a flat round, about 4 inches in diameter. Using a floured rolling pin, roll out dough on a lightly floured surface to a 7-inch round; transfer to a floured pizza peel.
3. Spread about 1/4 cup sauce onto dough, leaving a 1/4-inch border at edge. Working quickly, sprinkle with some of garlic, then add 1/6 of the mozzarella.Top with 1⁄6 of the spinach and 1⁄6 of the mushrooms. If desired, lightly drizzle pizza with additional oil and sprinkle with salt. Slide pizza off of peel onto stone and bake until edges are puffed and golden, about 5-6 minutes.
4. Transfer pizza to a cutting board; cut into 6 slices and serve immediately, with red pepper flakes, if desired. Repeat with remaining ingredients.