Heat garlic oil and butter in a skillet over medium heat. Add mushrooms; cook and stir until dark and tender, about 5 minutes.
Place tortilla on a pizza pan or baking stone. Spoon mushrooms, with accumulated juices, on top. Scatter mozzarella cheese and cherry tomatoes over mushrooms. Sprinkle oregano and garlic on top. Drizzle olive oil on top.
Bake in the preheated oven until edges of the tortilla start to brown, 8 to 10 minutes.