Heat a large skillet over low heat and melt the butter. Add the onions and stir to coat. Cook for about 1 hour, stirring every few minutes, until they're deeply golden brown and caramelized. If the onions start to dry out, add a tablespoon of water as needed to keep them from burning.
While the onions cook, make the garlic aioli. In a small bowl, whisk together the mayo, garlic, Dijon mustard, Worcestershire sauce, and lemon juice. Cover and refrigerate until you're ready to assemble the burgers. The longer it sits, the more flavorful it becomes.
In a large bowl, combine the ground turkey, mayo, salt, pepper, garlic powder, and paprika. Mix until everything is evenly incorporated, then divide the mixture into 6 equal portions.
Heat a large skillet over medium-high heat and lightly coat it with oil. Place the turkey portions into the skillet and use a spatula to gently press each one into a patty.
Cook for about 7 minutes on the first side, then flip and cook for another 3 minutes, or until fully cooked. During the last few minutes of cooking, place a slice of cheese on each burger and let it melt.
To assemble, place a lettuce leaf on a plate, followed by the turkey burger. Top with the garlic aioli, sliced tomato, and caramelized onions. Add any other favorite burger toppings and serve immediately.