Combine the turkey, garlic, paprika, 3/4 teaspoon salt and a few grinds of pepper in a medium bowl; mix with your hands until just combined. Form into four 1/2-inch-thick patties; set aside.
Combine 1/2 avocado, the mayonnaise, lime juice and 1/4 teaspoon salt in a medium bowl; mash until smooth. Thinly slice the remaining 1/2 avocado and season with salt and pepper. Set aside for topping.
Heat the vegetable oil in a large nonstick skillet over medium-high heat (or preheat a grill to medium high and oil the grates). Add the burgers and cook until browned, 3 to 4 minutes per side. Top each burger with a slice of cheese; cover and cook until the cheese melts, about 30 seconds.
Meanwhile, split and toast the buns; spread the avocado mayonnaise on the cut sides. Serve the burgers on the buns with lettuce, the sliced avocado and tortilla chips.