Slice the pepper into 2 or 3 large pieces. Place on a baking sheet, skin side facing up, and put in an oven heated to 450 degrees. Roast until the skin of the peppers is slightly charred, this will take about 15 minutes.
While the pepper is roasting, slice the chicken into strips ¼ inch thick, and sprinkle with half the salt and pepper. Chop the garlic. Heat ½ tablespoon olive oil in a skillet over medium heat. Add the garlic, and after 1 minute, add the chicken. Cook the chicken for 2 minutes per side, until no longer pink in the center. Then remove from the skillet and set aside.
Add the remaining olive oil to the skillet used to cook the chicken. Add the mushrooms and remaining salt and pepper. Use a spatula to scrape up any brown bits from the chicken and stir with the mushrooms to add flavor. Let sauté until slightly browned, this will take about 6 minutes.
Remove the pepper from the oven. Turn the oven off, but put the tortillas in the oven to warm while you peel the peppers. Let cool, and then use your fingers to peel the skin off the pepper. It doesn't have to be perfect, just remove any charred bits. Then slice the peppers into thin strips about ⅛ inch thick.
Loosely chop the cilantro. Remove the tortillas from the oven. Top the tortillas with the chicken, sautéed mushrooms, roasted poblano peppers, queso fresco, and cilantro.