Preheat oven and a ceramic pizza stone to 500°.
Roll, stretch and toss the dough into a thin, round shape, using plenty of flour to keep it from sticking. Place the dough on a sheet of parchment paper on top of a large cutting board.
In a medium frying pan, melt butter and add garlic, then quickly sauté chanterelles in the pan over medium-high heat until almost cooked through. Remove from heat.
Spread mozzarella, then the sautéed chanterelles, and the smoked salmon, evenly on the dough. Place a few thin rings of red onion sparingly on top. Sprinkle with Parmesan.
Place in oven on hot pizza stone and bake until golden brown and bubbling, about 12 to 15 minutes.