Preheat the broiler and char the poblanos all over until the skin is blackened, 12 to 15 minutes. Place in a bowl and cover to steam for about 10 minutes. Peel, seed and chop the chiles.
In a large skillet, heat 3 tablespoons EVOO, 3 turns of the pan, over medium-high heat. Add the mushrooms and cook until softened, about 10 minutes. Add the remaining 2 tablespoons EVOO, 2 turns of the pan, the onion and garlic and cook for 5 minutes. Add the hominy, cumin and oregano and heat through; season with salt and pepper. Stir in the tomato paste for 1 minute. Stir in the beer and cook until reduced, 1 minute. Stir in the poblanos.
Serve the posole in shallow bowls with the cabbage, cheese and lime wedges, and the tortillas for wrapping and eating.