Prick the chicken all over with a fork and put in a medium bowl. Rub the cumin and 3/4 teaspoon salt 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 avocado, remaining 1/4 cup sour cream, remaining lime zest and juice, 3 tablespoons water and 1/2 teaspoon salt in a blender until very smooth; refrigerate until ready to use.
Meanwhile, preheat the broiler. Line a rimmed 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, turning once, 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 half of the scallions to the bowl with the poblanos and toss to combine. Combine the cheese and garlic in a small bowl.
Top 4 tortillas with 1/4 cup of the cheese mixture, one-quarter of the chicken-poblano mixture then 1/4 cup more 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, turning once, 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 quesadilla into quarters and sprinkle with the remaining scallions. Serve with a dollop of the avocado cream, lime wedges and salsa on the side.
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