Pour 3/4 cup warm water (105–115°F) into a large bowl. Sprinkle 1 1/2 teaspoons dry yeast over the water, stir briefly, and let stand 10 minutes until the yeast has dissolved and the surface looks slightly foamy.
Add 1 tablespoon extra-virgin olive oil, 1 teaspoon salt, 1 teaspoon whole fennel seeds, 1 teaspoon dried thyme, and 1 3/4 cups (about) unbleached all-purpose flour to the bowl. Stir until a sticky dough forms.
Turn the dough onto a generously floured work surface. Knead until smooth and elastic, about 5 minutes, adding only enough flour to keep the dough from sticking; the dough should remain soft.
Shape the dough into a ball, place it in a large lightly oiled bowl, and turn to coat the surface of the dough. Cover the bowl with a kitchen towel and let the dough rise at cool room temperature until it is almost doubled, about 1 1/2 to 2 hours.
After the dough has risen, punch it down and form it into a ball. Return it to the bowl, cover with the towel, and let it rest 30 minutes. While the dough is resting, preheat the oven to 425°F.
Mix 1/4 cup olive oil, 3 minced garlic cloves, and 1 teaspoon red pepper flakes in a small bowl; set this garlic oil aside to infuse.
Lightly flour a rimless baking sheet. Turn the rested dough out onto a floured surface, roll or stretch it to a 13-inch round, and transfer it to the prepared baking sheet.
Bake the crust on the middle oven rack for 5–7 minutes, until it is just set (not fully browned).
Remove the par-baked crust from the oven and brush it evenly with 2 tablespoons of the garlic oil mixture.
Top the crust in this order: 2 cups (packed) baby spinach leaves, 1 1/2 cups thickly sliced mushrooms, 1/2 cup drained roasted red peppers (cut into thin strips), 1/2 cup paper-thin red onion slices, 8 large fresh basil leaves (cut into thin strips), and the 5-ounce package of crumbled soft fresh goat cheese.
Drizzle the remaining garlic oil evenly over the assembled pizza.
Return the pizza to the oven and bake 15–18 minutes, until the crust is crisp and the goat cheese begins to brown.
Transfer the pizza to a cutting board, cut into wedges, and serve warm.