Preheat the broiler. Place the chiles on a rimmed baking sheet and broil, turning occasionally, until charred on all sides, about 10 minutes. Transfer to a bowl, cover and let stand until cool enough to handle; remove and discard the skins. Cut a slit lengthwise in each chile and discard the seeds, leaving the stems attached and being careful not to tear the chiles.
In a large nonstick skillet, heat 1 tablespoon olive oil over medium heat. Add the mushrooms and cook, stirring, until browned, about 5 minutes. Add the onion and half of the garlic and cook, stirring, for 1 minute. Stir in the tomatoes and cook until the onions are softened, about 5 minutes. Stir in the cheese; season with salt and pepper. Let cool slightly.
Preheat the oven to 350 degrees . Stuff the chiles with the vegetable-cheese mixture and place in a baking dish. Cover the dish and bake until heated through, about 15 minutes.
Meanwhile, in a small skillet, heat the remaining 1 tablespoon olive oil over medium heat. Add the remaining garlic and cook, stirring, until golden, about 2 minutes. Stir in the refried beans and 1/3 cup water and cook until warmed through; season with salt and pepper.
Serve the stuffed chiles with the refried beans and yogurt, if using.