Sauté the onions in a pan with the olive oil over medium-low heat for about 15 minutes until the onions are glossy, soft, and browned. Stir about once a minute and watch constantly to keep them from burning. Set aside.
Mix the turkey with the remaining ingredients (except for the buns) and form the mixture into patties.
Grill or cook in a skillet with a little oil until the center is cooked through. Well done is best.
Place the burgers on the bottom buns, top with caramelized onions, and cover with the top buns.