How to Make It
Combine the cucumber, radishes, lemon juice, dill, 2 tablespoons of the oil, and ¼ teaspoon each salt and pepper in a medium bowl.
Gently mix together the turkey, barbecue sauce, ½ teaspoon salt, and ¼ teaspoon pepper with your hands in a large bowl until just combined. Dividing evenly, form the turkey mixture into four ¾-inch patties. Use your fingers to make a shallow well in the top of each patty (this will prevent overplumping during cooking).
Heat the remaining tablespoon of oil in a large nonstick skillet over medium heat. Add the patties and cook until cooked through and an instant-read thermometer inserted in the center registers 165° F, 4 to 6 minutes per side.
Top the burgers with the Cheddar, dividing evenly, and cook, covered, until the Cheddar is melted, 1 to 2 minutes.
Stack the lettuce, burgers, and additional barbecue sauce between the buns. Serve with the cucumber salad and potato chips.