Smash the garlic cloves, sprinkle with some salt, and, with the flat side of a large knife, mash and smear the mixture to a coarse paste. Whisk the garlic with the balsamic, Worcestershire, and hot sauce in a small bowl.
Crumble the turkey and beef into a large mixing bowl. Pour the garlic mixture over the meat and form into 4 (5-ounce) burgers. Cover and set aside while you cook the onions.
Meanwhile, heat 1 tablespoon of the olive oil in a skillet over medium-high heat. Add the onion and cook, stirring occasionally, until browned but not mushy, about 15 minutes. Season the onions with the remaining teaspoon of balsamic and salt and pepper, to taste.
Heat a large nonstick skillet over medium-high heat for a minute. Brush 1 side of the burgers with the oil and season with salt and pepper, to taste. Cook seasoned side down until browned, about 2 minutes. Brush and season the top of the burgers, flip and cook until other side is brown, about 2 minutes more.
Transfer burgers to a baking dish and bake until an instant-read thermometer inserted sideways into the burgers registers 160 degrees F, about 5 minutes. Place the burgers on the buns or in pita and let rest for a few minutes. Serve with the caramelized onions and garnishes of your choice.