Preheat the oven to 400°F. Gently stir together Monterey Jack, Muenster, and queso fresco in a large bowl.

Spread chiles in an even layer on a rimmed baking sheet, and drizzle with 1 tablespoon of the oil. Bakein the preheated oven until chiles have softened and skins have blistered, 20 to 25 minutes. Remove chiles from oven; do not turn off the oven. Peel and seed chiles under cool running water. Chop chiles, and set aside.

Brush a 12-inch cast-iron skillet with 1 tablespoon of the oil, and add cheese mixture to skillet. Bake until cheese has melted and top is golden brown, 20 to 25 minutes.

Meanwhile, heat remaining 2 tablespoons olive oil in a large sauté pan or skillet over medium. Add garlic, and cook, stirring often, until slightly browned, about 1 minute. Add mushrooms, and cook, stirring often, until mushrooms have softened and liquid has almost completely evaporated, 6 to 7 minutes. Add roasted chiles, and cook, stirring often, until mixture is hot, 1 to 2 minutes.

Carefully remove the hot skillet from oven. Top cheese dip with mushroom mixture. Serve with tortilla chips.