These portobellos are stuffed with garlic-and-Parmesan-infused mashed potatoes, making a hearty vegetarian dish.
Heat oven to 400 degrees. Wrap garlic cloves in aluminum foil and bake until garlic is tender, 25 to 30 minutes. Remove from oven and set aside. When cool enough to handle, squeeze roasted garlic from skins and set aside. Leave oven at 400 degrees.
Scrub potatoes and place in a large pot of cold water. Bring water to a boil, add 3 teaspoons salt, and reduce heat. Gently boil until potatoes are tender, about 40 to 50 minutes. Remove from heat and drain. Hold hot potatoes in a clean dish towel and peel.
While potatoes are boiling, warm milk in a small saucepan over medium-low heat until bubbles form around the edges.
Bring a large pot of water to a simmer. Using a ricer or a food mill, puree potatoes into a large metal bowl, then place bowl over the pot of simmering water. Add the roasted garlic, warm milk, Parmesan, butter, 1 teaspoon salt, and 1/4 teaspoon pepper to potatoes and stir until smooth and creamy. Cover bowl with aluminum foil and keep warm over the barely simmering water until ready to use.
Meanwhile, wipe mushrooms with a cloth or soft brush to remove dirt. Remove and discard mushroom stems. Place mushroom caps in a large roasting pan, stem side down, and drizzle with 1 1/2 teaspoons olive oil. Season with 1/2 teaspoon salt and 1/8 teaspoon pepper. Turn and drizzle with remaining 1 1/2 teaspoons olive oil, and sprinkle with remaining 1/2 teaspoon salt, 1/8 teaspoon pepper, and thyme leaves. Place pan in oven and bake mushrooms until tender, about 15 to 20 minutes.
Remove mushrooms from oven and heat broiler. Spoon about 1/2 cup potato puree onto each mushroom.
Place under broiler until potatoes are golden brown, about 2 to 3 minutes and serve immediately.
This dish can be assembled several hours ahead through step 6. Reheat in a 350-degree oven for 30 minutes before broiling.
Yukon Gold potatoes can be replaced with Yellow Finn potatoes.