To prepare pizza dough, lightly flour a pizza peel and gently stretch dough into a thin rectangle, making sure not to get dough too thin. Brush oil over pizza dough, leaving a 1-inch border around edges.
In a medium sauté pan over medium heat, melt 1 tablespoon butter. Add onions and sauté until onions are caramelized, stirringly occasionally, about 10-15 minutes. Season with salt and pepper. Transfer to a plate and set aside.
Melt remaining butter and add mushrooms. Stir until just cooked through, about 5-7 minutes. Stir in thyme and cook for another 1-2 minutes. Set aside.
Place fontina cheese on bottom of dough, followed by mushrooms. Top with onions and Parmigiano-Reggiano and season with salt and pepper.
Gently dust cornmeal onto hot pizza stone. Transfer dough to pizza stone and cook about 10-12 minutes, until crust gets lightly browned and cheeses are just melted.
Remove pizza from oven and cut into slices. Serve warm.