1Heat 1 tbsp vegetable oil in a large skillet over medium-high heat. Add chicken pieces and cook until cooked through, about 8–10 minutes. Season with cumin, chili powder, salt, and pepper. Transfer to a plate.
2In the same skillet, add remaining 1 tbsp oil. Sauté onion, poblano, and bell pepper until soft and lightly charred, about 6 minutes. Add garlic and cook 1 minute more. Return chicken to the skillet and toss to combine. Remove from heat.
3Wipe the skillet clean. Working one at a time, lay a tortilla flat in the skillet over medium heat. Scatter a handful of oaxaca cheese on one half, top with about 2–3 tbsp of the chicken mixture and a pinch of cotija. Fold in half, press gently, and cook until golden and crispy, about 2 minutes per side. Repeat with remaining tortillas.
4For the lime crema: whisk sour cream with lime juice and a pinch of salt until smooth.
5Serve quesadillas hot with lime crema, salsa, pickled jalapeños, and fresh cilantro on the side.