Preheat oven to 475 degrees. On a parchment-lined baking sheet, rub peppers with 2 teaspoons oil. Roast until slightly blackened and softened, 15 to 20 minutes, turning once. When cool enough to handle, remove skins. Make a small slit in each pepper and carefully remove seeds, keeping stem end intact.
In a blender, combine chipotle, garlic, and 3/4 cup water; season with salt and pepper and blend until smooth. Add cilantro and pulse once to combine.
In a small saucepan, bring 2/3 cup water to a boil. Add quinoa, cover, and reduce heat to medium-low. Cook until tender, about 10 minutes. Remove from heat and let stand 5 minutes; fluff with a fork.
Meanwhile, in a large skillet, heat 1 tablespoon oil over medium-high. Add mushrooms; season with salt and pepper. Cook until mushrooms are tender, about 5 minutes; add corn and beans and cook until heated, about 2 minutes. Stir in 1/2 cup cheese and quinoa.
Pour sauce in a 9-by-13-inch baking dish. Stuff peppers with filling and arrange in dish; dot with 1/2 cup cheese. Bake until golden, 15 to 20 minutes. Cool in pan 10 minutes.