2 large poblano peppers
4 boneless, skinless chicken breasts
Salt and black pepper
3 tablespoons extra virgin olive oil (EVOO), divided
1 onion, chopped
2 cloves garlic, finely chopped
8 medium tomatillos, rinsed and chopped
1/2 cup cilantro, leaves removed from stems (approximately 1 small handful)
Juice of 1 lime
2 teaspoons honey
12 8-inch flour tortillas
1 1/2-2 cups shredded Swiss cheese
1 1/2-2 cups shredded Monterey Jack cheese
Heat the broiler to high. Place the poblanos under the broiler and char until blackened on all sides, 10-12 minutes. Leave the door of the oven cracked to allow the steam to escape. Place the peppers in a bowl and cover tightly with plastic wrap. Allow the peppers to cool until cool enough to handle. Switch the oven to low, about 225°F. While the peppers char, place the chicken in a single layer in a small pan and add enough water to come up to the top of the meat but do not submerge completely; season with salt and pepper. Bring to a boil, then reduce to simmer and poach for 10-12 minutes, or until cooked through. Flip the chicken breasts during the last 4 minutes of cooking.
While the chicken is poaching, prepare the Suiza sauce: heat 2 tablespoons EVOO, two turns of the pan, in a skillet over medium heat. Add the onion and garlic and cook to soften, 5 minutes.
While the onion and garlic cook, place the tomatillos, cilantro and lime juice in a food processor. Peel and seed the poblano peppers and add them to the food processor, along with the onion garlic mixture. Process until an almost smooth sauce forms, about 30 seconds. Pour the sauce into a skillet, add in the honey, season with salt and pepper and simmer over low heat for about 10 minutes, stirring every couple minutes.
Shred the cooked, poached chicken with two forks and reserve. Pour the Suiza sauce over the chicken and mix to combine.
Spread some Swiss and Monterey Jack cheese on two of the tortillas. Place a large handful of the chicken mixture on one tortilla and spread to evenly coat. Make quesadilla stacks, starting with a cheese topped one, then chicken, then cheese and top with a plain tortilla to seal it up. Repeat with the remaining tortillas, alternating the filling the same way as before and topping with a plain tortilla to make three Quesadilla Suiza Stacks.
Pre-heat a skillet over medium-low heat with the remaining tablespoon of EVOO and place one stack into the skillet. Cook until brown on each side, about 2-3 minutes per side. When the first stack is done, pop it into the oven to stay warm while you brown the others.
To serve, cut each Quesadilla Suiza Stack into quarters using a serrated knife and serve with Avocado Sour Cream alongside.