Preheat oven to 375°F and place rack in lower third of oven. Heat oil in a large skillet over high heat. Add garlic and onions and cook, stirring constantly, for 2 to 3 minutes, until garlic begins to brown. Add sausage and cook for 10 to 12 minutes, until onions are translucent and sausage is fully cooked. Remove from heat and stir in bread crumbs, parsley, and salt and pepper. Allow to cool.

Meanwhile, remove stems from mushrooms and, using a small spoon, scrape brown gills out of caps. (This step prevents any discoloration in the final dish.) Brush each cap well with olive oil, then arrange them gill-side up on a rimmed baking sheet. Layer caps with spinach, creating a bed for the filling. Divide sausage mixture equally among mushroom caps and press into a neat mound. Sprinkle cheese on top of filling and bake until cheese is crisp and brown, 20 to 30 minutes. Transfer to plates and serve.