For filling, in a medium skillet cook chorizo or sausage, onion, and garlic until meat is brown and onion is tender. Drain off fat. Pat chorizo mixture with paper towels to remove as much additional fat as possible. Stir in jalapeno pepper; set aside.
Brush one side of 3 tortillas with half of the cooking oil. Place tortillas, oiled sides down, on a large baking sheet. Spread the chorizo filling over tortillas on baking sheet. Combine cheese and cilantro or parsley; sprinkle over filling. Top with the remaining tortillas. Brush with the remaining oil.
Place quesadillas on the rack of an uncovered grill directly over medium heat. Grill for 3 to 4 minutes or until filling is heated through and tortillas are starting to brown, turning once halfway through grilling. To serve, cut quesadillas into wedges. Makes 8 servings.