Heat oven to 375° F. Remove the stems from the mushrooms; finely chop the stems and reserve. Coat a baking sheet with oil. Bake the mushroom caps, stem-side down, until just tender, 10 to 12 minutes.
Meanwhile, heat the oil in a medium skillet over medium heat. Add the garlic and chopped mushroom stems and cook, stirring, until tender, 3 to 5 minutes. Add the spinach, ¼ teaspoon salt, and ⅛ teaspoon pepper and cook, tossing, until wilted, 2 to 3 minutes more.
In a bowl, combine the spinach mixture with the bread crumbs and cheese. Spoon into the mushroom caps and bake until golden brown, 12 to 15 minutes.