Cover chicken with salted water in a large pot. Bring just to a boil, then immediately reduce to a gentle simmer. Cook for about 15 to 20 minutes or until the chicken is cooked through. Remove chicken to a large pan or bowl (reserve the broth) and allow chicken to cool a little. Shred chicken into medium-to-large bite-size pieces; cover and set aside.
Meanwhile, line a baking sheet with parchment paper. Place poblano chiles on the sheet and brush with 2 teaspoons of the olive oil. Roast in a 400-degree oven until chile skins begin to blacken, turning the chiles until blistered on all sides. Remove from oven and place in a paper bag to cool. When cooled, slip the blackened skins off the chiles, removing and discarding skins and stems.
While chiles roast and cool, saute onion and garlic over medium heat in remaining tablespoon olive oil until softened, about 5 to 7 minutes. Stir in cumin until well incorporated and remove from heat.
In a blender, combine chiles with seasoned onion and garlic. Add cilantro leaves and enough of the reserved chicken broth to make the mixture smooth while pulsing. Add heavy cream and salt to taste, pulse again a few times, adding a little more cream as needed to make a sauce that’s just slightly thick; don’t let the sauce become watery.
Tear tortillas into four pieces each.
Butter a large, rectangular casserole dish (10 x 14 or 15 is best, but 9 x 13 will do). Cover the bottom of the casserole dish with a few spoonfuls of the poblano cream sauce. Build layers starting with torn tortillas, shredded chicken, sauce, and mixed cheeses. Repeat the layering twice more, covering the top with grated cheese.
Bake in a 350-degree oven for 35 to 45 minutes or until bubbly and browned at the edges. Remove from oven and allow to set for 10 minutes before serving. Consider serving with a bowl of salsa verde, too.