At least 1 hour before baking, heat the oven to 550°F with a baking steel or stone on the upper-middle rack and a second rack in the lower-middle.

Using a spoon, scrape off and discard the gills on the undersides of the mushroom caps. Halve any caps that are 5 inches or larger in diameter, then cut the caps into ¼-inch slices. In a large bowl, toss the mushrooms, olive oil and ¼ teaspoon salt.

Spread the mushrooms in an even layer on a rimmed baking sheet. Roast on the lower oven rack, stirring once, until they have released their moisture, the moisture evaporates and the mushrooms begin to brown, about 15 minutes.

Stir in the thyme and garlic, then roast until the mushrooms have browned and the garlic is no longer raw, another 3 to 4 minutes. Let cool completely on a wire rack. Leave the oven on.

Turn the dough out onto a counter dusted generously with bread flour. Flour your hands and, using your fingers, press the dough, starting at the center and working out to the edges, into a 12-inch round, turning the dough over once.

The round should be thin in the center, with slightly thicker edges. Lightly dust a baking peel, inverted baking sheet or rimless cookie sheet with the semolina. Transfer the dough to the peel and, if needed, reshape into a 12-inch round.

Using the back of a spoon, spread the fontina-Parmesan cream evenly on the dough, leaving ½-inch border at the edge.

Scatter the mushrooms over it and season with pepper. Slide the pizza onto the baking steel or stone and bake until the crust is well browned, 7 to 9 minutes.

Using the peel, transfer the pizza to a wire rack. Let cool for a couple of minutes, then sprinkle with scallions and red pepper flakes.