How to Make It
Put yeast in a small bowl; add water and sugar. Gently whisk to combine. Let stand until foamy, about 5 minutes.
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.
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.
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.
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.
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.
Transfer pizza to a cutting board; cut into 6 slices and serve immediately, with red pepper flakes, if desired. Repeat with remaining ingredients.