Heat olive oil in a large skillet over medium heat. Add peppers and reduce heat to low.
Cook peppers until they soften and just begin to caramelize. Stir in sugar and vinegar.
Cook an additional minute or two or just until liquid evaporates. Set aside
Heat grill. In a bowl, combine turkey, onions, breadcrumbs, egg, liquid smoke and Worcestershire sauce. Sprinkle with salt and pepper.
Using your hands, combine turkey mixture until well blended. Shape into 4 patties.
Dip each patty in the panko crumbs and grill burgers 4 minutes on each side or until a thermometer registers 165°F. Divide peppers among burgers, top with cheese and heat until cheese has melted.
In a small bowl, combine mayo and garlic. Cover tops and bottoms of each bun with mayo.