Cook all potatoes, garlic, and bay leaf in large saucepan of boiling salted water until very tender, about 12 minutes. Drain; discard bay leaf. Return potatoes to pot. Mash until smooth; cool slightly. Mix in 3 tablespoons cheese. Season with salt and pepper. Whisk in eggs.
Remove mushroom stems and chop finely. Heat 2 teaspoons oil in large nonstick skillet over medium-high heat. Add mushroom stems and onion; sauté 5 minutes. Stir in parsley and rosemary. Season with salt and pepper.
Preheat oven to 400°F. Brush rounded sides of mushrooms with 3 teaspoons oil. Place on baking sheet, gill sides up. Bake 10 minutes. Reduce oven temperature to 350°F. Divide onion mixture among mushroom caps. Mound potato mixture in caps. Sprinkle with 1 tablespoon cheese. Bake until heated through, about 25 minutes.
Heat 1 teaspoon oil in large pot over medium heat. Add spinach; sauté 2 minutes. Season with salt and pepper. Divide among 4 plates. Top each serving with 1 stuffed mushroom.
