Pizza with Five Onions and Wild Mushrooms

Ingredients:

3 tablespoons unsalted butter, divided
2 tablespoons olive oil, divided
1 red onion, thinly sliced
1 yellow onion, thinly sliced
2 leeks, white and light-green parts only, cut in half lengthwise, thinly sliced, ]well-rinsed
4 scallions, roughly chopped
Kosher or sea salt and freshly ground black pepper
2 shallots, thinly sliced
1/4 pound wild mushrooms, roughly chopped
1/2 cup dry white wine (or chicken stock)
1 sprig fresh rosemary (or 3 to 4 sprigs fresh thyme)
1 pound pizza dough, divided
Flour
1 cup grated Fontina or crumbled goat cheese (or your favorite cheese), divided