In a mixing bowl, combine mayonnaise, minced garlic, lemon juice, and salt. Mix well and set aside.

Heat olive oil in a skillet over medium heat. Add sliced onions and cook, stirring occasionally, for about 15 minutes until they are golden brown and caramelized. Add balsamic vinegar if desired and stir. Remove from heat.

In a mixing bowl, combine ground turkey, garlic powder, onion powder, salt, and black pepper. Mix until just combined. Form into 4 patties.

Preheat the grill to medium-high heat. Cook the turkey patties for about 5-7 minutes on each side or until they reach an internal temperature of 165°F (74°C). If using cheese, add it on top of the patties during the last minute of cooking to melt.

Spread aioli on the bottom half of each toasted bun. Place a turkey patty on each bun, top with caramelized onions, lettuce, and tomato. Cover with the top half of the bun and serve.