For Peach Ketchup, in a blender or food processor combine cut-up peaches, sugar, cider vinegar, chili powder, cinnamon, and cayenne pepper. Cover and blend or process until smooth.
In a large bowl lightly mix turkey with half of the Peach Ketchup, the salt, and black pepper. Divide into eight balls. Make an indentation in the center of each ball and fill with some of the cheese. Shape meat around cheese. Flatten into patties.
In a 12-inch skillet heat oil over medium heat. Add half of the burgers and cook about 5 minutes or until well browned. Turn and cook 5 minutes more or until no pink remains (165 degrees F). Repeat with remaining burgers.
Serve on lettuce-lined buns with additional peach slices and remaining Peach Ketchup.