Make the Thai-style sauce: in a saucepan whisk together sweet chili sauce, rice vinegar, coconut milk, brown sugar, 3 cloves minced garlic, grated ginger, peanut butter, soy sauce, and lime juice until smooth.
Bring the sauce to a gentle boil over medium heat, let it bubble for 1 minute, then reduce heat to low and simmer for 2–3 minutes; remove from heat and let cool.
Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Season the cubed chicken with salt, pepper, and cumin, then add to the skillet and cook, stirring occasionally, until browned and cooked through, about 5–6 minutes; transfer to a plate.
Add remaining 1/2 tablespoon oil to the same skillet. Add diced onion, shredded Napa cabbage, chopped carrots, and 1 minced garlic clove; toss to coat and cook until slightly wilted but still crisp, about 3–4 minutes.
Return the cooked chicken to the skillet with the vegetables and stir to combine, then remove from heat.
Assemble each quesadilla on a tortilla: sprinkle a little cheese on the tortilla to help bind, add half of the chicken-vegetable mixture, scatter chopped peanuts and chopped cilantro, spoon 2–3 tablespoons of the Thai sauce over the filling, and top with more cheese; place a second tortilla on top.
Cook the assembled quesadilla in the skillet or on a griddle over medium heat until the bottom is browned and crisp and the cheese begins to melt, then carefully flip and cook the other side until golden and cheese is fully melted; repeat for the second quesadilla.
Slice each quesadilla into wedges and serve with the remaining sauce for dipping.
