Preheat the oven to 550°F or as high as it will go (if using a stone/steel, put it on a rack in the center or bottom third of the oven before you start preheating).
Heat the oil in a medium skillet over medium-high heat. Add the garlic, cook until it is aromatic, about 1 minute, then add the mushrooms. Cook, stirring, until the mushrooms release their liquid, about 5 minutes, then add the spinach and cook, stirring, just until the spinach is wilted. Remove the skillet from the heat and set aside.
Stretch or roll out your dough to a 12-inch circle, then transfer it to an oiled baking sheet or a lightly floured pizza peel (if using a baking stone/steel).
Spread the marinara sauce evenly over the dough, leaving a 1/2-inch border all around. Top with the spinach-and-mushroom mixture and finish with the shredded Swiss cheese. If you like (or if your diet allows), brush the edges of the dough with olive oil.
Transfer the pizza to the oven and bake until the cheese has browned in spots, 8 to 10 minutes on the baking sheet, 6 to 8 minutes on the baking stone/steel.
Take the pizza out of the oven. Let it cool for a few minutes before slicing and serving.