In a large saute pan over medium heat, melt 2 tablespoons butter with 1 tablespoon olive oil. Add onions, leeks, and scallions; sweat, stirring often, until softened and silky, about 45 minutes. Season with salt and pepper to taste. Set aside and let cool to room temperature. In a large saute pan over medium-high heat, melt remaining tablespoon of butter with remaining tablespoon of olive oil. Saute shallots and mushrooms, stirring often, about 8 minutes. Add wine (or chicken stock) and rosemary (or thyme) and simmer until almost all liquid is absorbed, about 10 minutes. Season with salt and pepper to taste. Remove herb sprigs, and cool to room temperature. Heat oven to 500 degrees. Cut pizza dough in half. On lightly floured surface, roll one piece of dough out into a thin, 12-inch round. Sprinkle a rimless baking sheet (or flip over a rimmed baking sheet) with flour. Transfer dough round to baking sheet. Scatter half the Fontina cheese evenly onto dough. (If you’re using goat cheese or another soft cheese, sprinkle on top as the last step.) Scatter half the onion mixture over the cheese. Scatter half the mushroom mixture over onions. Repeat with remaining ingredients. Bake 8 to 10 minutes, or until edges are well browned.
While you’re slicing all those onions, be sure to saute some extra and freeze them for your next pizza adventure–or add them to sauces, scrambled eggs, or soups.