Make the buttermilk marinade: In a medium bowl, combine the buttermilk, egg, salt, pepper, and cayenne. Toss in the chicken thighs in the marinade to coat. Marinate for at least 1 hour in the refrigerator, or overnight.
Make the dill dressing: In a small bowl, combine the yogurt, dill, garlic powder, lemon juice, and Parmesan cheese. Cover and let sit for at least 1 hour in the refrigerator to chill.
Make the seasoned flour: In a medium bowl, combine the flour, salt, pepper, cayenne, and garlic powder. Dip the marinated chicken in the flour mixture until the chicken is completely covered.
Heat the oil in a deep pot until it reaches 350˚F (180˚C). Do not fill more than halfway with oil. Carefully fry the chicken for 7 minutes, or until cooked through.
The internal temperature should reach 165˚F (75˚C), and the chicken should be golden brown and crispy. Drain on a paper towel–lined plate or wire rack.
Heat a large skillet over medium heat. Butter the cut sides of the burger buns, then toast on the hot skillet until browned and crisp. Spread mayonnaise mixture on bottom halves of buns.
Build the sandwiches with the toasted buns, lettuce, fried chicken, tomato slices, and dill dressing.