Preheat oven to 350°F. Heat 1 tablespoon of oil in a large skillet over medium-high heat. Add mushroom stems, onions, salt and pepper and cook, stirring occasionally, until softened, 8 to 10 minutes. Transfer to a large bowl along with spinach, toss well; set aside to cool. Arrange mushroom caps in a single layer on a parchment paper-lined baking sheet. Add Feta, salt and pepper to spinach mixture then divide filling evenly between mushrooms, mounding it in the center of each. Drizzle mushrooms with remaining 1 tablespoon oil and bake until softened and juicy, about 20 minutes.