Melt 2 tablespoons butter in a large skillet over medium heat. Place the chicken breasts in the skillet. Mix in the poblano peppers, jalapeno peppers, and garlic. Squeeze the juice from one lime half over the chicken, and place the lime peel and pulp into the skillet. Cover, and simmer for 25 minutes, stirring occasionally, until chicken juices run clear. Remove from heat, cool, and cut chicken into cubes.

Preheat oven to 350°F (175°C).

Melt 1/4 cup butter in a medium saucepan over low heat. Gradually stir in the flour and half-and-half. Mix in the cilantro, mushrooms, sour cream, and juice of the remaining lime half. Cook and stir for 10 minutes, until mushrooms are tender.

Line the bottom of a 9x13 inch baking dish with 6 tortillas. Arrange 1/2 the chicken and pepper mixture over the tortillas. Sprinkle with 1 cup cheese, and cover with remaining tortillas. Layer with the remaining chicken and pepper mixture, and pour the cilantro sauce evenly over the casserole. Top with remaining cheese.