Preheat the oven to 375 degrees . In a large skillet, heat the olive oil over medium-high heat. Add the chopped mushroom stems and cook until softened, 5 to 7 minutes; season with salt. Add the sausage and garlic, breaking up the meat, and cook until no longer pink, about 5 minutes.
Transfer the mixture to a bowl. Add three-quarters of the cheese and arugula; season with salt and pepper. Divide the mixture among the mushroom caps and place on a baking sheet. Sprinkle with the remaining cheese and bake until the caps are softened, 12 to 14 minutes. Top with the remaining arugula.