Place each turkey burger between two pieces of waxed paper. Press to 3/8-inch thickness. Place 1 ounce cheese on half of each patty to within 1/2 inch of edge. Fold other half of patty over cheese; press edges to seal. Refrigerate at least 1 hour.
Melt butter in medium skillet over medium-high heat. Add onions; cook and stir 15 minutes or until golden brown. Stir in gravy. Reduce heat; simmer until heated through.
Coat large skillet with nonstick cooking spray. Heat over medium heat. Place burgers in skillet; cook 4 to 5 minutes per side or until no longer pink and center of burger reaches 165°F as measured with a meat thermometer. Serve with onion gravy.