Place olive oil, basil, chives and salt and pepper in a food processor and process until smooth.
Prepare a grill or coals.
Lay 4 tortillas out on a work surface and top each with cheese and onion. Sprinkle each with salt and pepper. Top with another tortilla and brush with oil.
Grill oiled side down and brush top side with oil. Turn when golden and grill other side until golden and cheese melts.
Remove and cut each quesadilla into quarters. Top each quarter with a slice of prosciutto and drizzle with basil oil.