Bloom the yeast: In a large bowl, combine the warm water, active dry yeast, and granulated sugar. Let stand until foamy, 5–8 minutes.
Make the dough: Stir in 2.5 cups of the bread flour, 1.5 tsp of the fine sea salt, and 1 tbsp of the extra-virgin olive oil until a shaggy dough forms. Knead on a lightly floured surface (use some of the remaining flour) until smooth and elastic, 8–10 minutes, adding just enough of the remaining flour to prevent sticking.
First rise: Lightly oil the bowl with a dab of the measured olive oil, place the dough inside, turn to coat, cover, and let rise at warm room temperature until doubled, 60–75 minutes.
Make the sauce: Warm 2 tbsp olive oil in a saucepan over medium heat. Add the grated garlic and cook until fragrant, about 30 seconds. Stir in the crushed tomatoes, remaining 0.5 tsp salt, dried oregano, dried basil, and black pepper. Simmer, stirring occasionally, until slightly thickened and bright, 15–20 minutes. Cool to room temperature.
Prepare toppings: Heat the remaining 1 tbsp olive oil in a skillet over medium-high heat. Add the Italian sausage and cook, breaking it into small crumbles, until browned and cooked through, 5–7 minutes; transfer to a plate. In the same skillet, add the sliced cremini mushrooms and cook until they release their moisture and turn golden at the edges, 4–6 minutes; set aside to cool.
Preheat: Place a pizza stone or steel on the middle rack and preheat the oven to 500–550°F for at least 45 minutes. If using an inverted sheet pan, preheat it as well.
Shape the dough: Gently deflate the risen dough and shape it into a tight ball. Rest 5 minutes. On a lightly floured surface, stretch or roll into a 14-inch round. Dust a pizza peel with semolina flour and transfer the dough to the peel.
Assemble: Spread an even layer of sauce over the dough, leaving a 0.5-inch border. Scatter the shredded low-moisture mozzarella over the sauce. Top with pepperoni, cooked sausage, sautéed mushrooms, green bell pepper, red onion, and black olives in an even layer.
Bake: Slide the pizza onto the preheated stone/steel and bake until the crust is deep golden and the cheese is bubbling with browned spots, 12–15 minutes, rotating once halfway if needed.
Finish and serve: Transfer to a cutting board, sprinkle with the finely grated parmesan, rest 2 minutes, then slice into 8 pieces and serve hot.