Heat 2 Tbl of the olive oil over high heat in a large sauté pan. When the oil becomes very hot, add mushrooms to the pan. Cook one minute without moving the mushrooms at all. Stir mushrooms after the first minute, and cook one more minute. Add the diced onion, garlic, Creole seasoning and black pepper and cook 5 more minutes, until the onions are tender and the moisture from the mushrooms has cooked out. Remove from the heat and place in a mixing bowl. Stir in the green onions, pesto and chicken and allow mixture to cool completely. Add the shredded cheese.
Lay the flour tortillas on a flat surface and divide the filling evenly among them. Spread the filling so that it covers half of the tortilla. Fold tortilla in half to form a semicircle.
Prepare the grill. Using a pastry brush, lightly coat both sides of the quesadilla with the remaining olive oil. Grill the quesadillas over direct medium heat until well marked, about 4-6 minutes on each side. Rotate the quesadillas while cooking to prevent burning.
Remove the quesadillas from the grill and cut them into 4-6 triangles.
Yield: 6-10 servings