In a medium mixing bowl, combine avocados with olive oil, lime juice, oregano, and salt and pepper, to taste. Divide mixture, reserving half to add to the ground turkey and half to use as a topping on the cooked burgers.
Melt butter in a medium skillet over low heat. Cook onions with salt to taste, stirring and shaking pan frequently, until onions are golden brown, about 15 minutes. Pour in vinegar, turn heat up to high, and reduce by half. Remove from heat and stir in cracked pepper (about 1 Tbsp.). Cool to room temperature.
Add turkey, caramelized onions, and olive oil to a large mixing bowl and combine thoroughly. Using hands, gently mix in half the avocados and form turkey mixture into 4 burger patties.
Preheat grill, broiler, or sauté pan over medium-high heat. Season turkey burger patties with salt and cook until lightly browned on bottom, about 3 to 4 minutes. Turn burgers and top with cheese slices. Continue cooking until turkey is just cooked through and cheese is melted.
Arrange buns, grilled or toasted side up, on plates. Place burgers on bottom halves of buns. Stir tomato into second half of reserved avocados and spoon mixture onto top halves of buns. Serve immediately.
Recipe adapted from: www.californiaavocado.com