Preheat oven to 425ºF. Coat a large skillet or wok with cooking spray; add oil and heat over high heat until sizzling.
Add chicken and 1 teaspoon of soy sauce to wok or skillet; cook, stir-frying, until well-browned, about 5 minutes. Add vegetables, scallions and remaining teaspoon of soy sauce; cook about 2 minutes. Add hoisin sauce; stir-fry until heated through, about 1 minute.
Place 4 tortillas on a large baking sheet (or 2 smaller baking sheets). Spread about a heaping 3/4 cup of chicken-vegetable mixture over each tortilla and sprinkle each with 2 tablespoons of cheese; top with remaining tortillas.
Bake quesadillas until golden brown and cheese melts, about 10 minutes. Remove from oven and let sit about 5 minutes before cutting each quesadilla into 4 wedges each. Yields 2 wedges per serving.