Poke holes all over the chicken with a fork, and put in a medium bowl. Rub 3/4 teaspoon salt and the cumin into the chicken. Add 1/4 cup of the sour cream and half the lime zest and juice, and mix until the chicken is coated. Let sit at room temperature for 15 minutes.
Blend the lime zest and juice, the avocado, remaining 1/4 cup sour cream, 3 tablespoons water and 1/2 teaspoon salt in a blender until very smooth; refrigerate until ready to use.
Meanwhile, preheat the broiler, and line a baking sheet with foil. Put the poblanos, skin-side up, on the prepared baking sheet, and broil until the skin is wrinkled and charred in spots, 2 to 3 minutes. Transfer the poblanos to a medium bowl, cover with plastic wrap and let steam for 5 minutes. Run the poblanos under cold water, and remove and discard the skin. Cut the poblanos into small bite-size pieces, and return them to the bowl.
Transfer the chicken to the baking sheet, and broil until browned in spots and cooked through, 4 to 5 minutes per side. Let cool for a few minutes. Cut it into small bite-size pieces. Add the chicken and 1/2 the scallions to the bowl with the poblanos, and toss to combine. Combine the cheese and garlic in a small bowl.
Lay out 4 of the tortillas. Top each with 1/4 cup of the cheese mixture, 1/4 of the chicken-poblano mixture, then 1/4 cup more of the cheese mixture. Top each with 1 of the remaining 4 tortillas, and press lightly.
Heat 1 tablespoon of the oil in a large nonstick skillet over medium heat. Add 1 quesadilla to the skillet and cook until it is golden and the cheese is melted, 2 to 3 minutes per side. Transfer to a baking sheet or large plate and repeat with the remaining oil and quesadillas. Cut each into quarters, and dollop with some of the avocado cream, salsa and remaining scallions.
Sponsor recipe courtesy of Sargento®