Place the chicken breasts between two sheets of plastic wrap or parchment paper. Using a meat mallet or rolling pin, pound the chicken to an even thickness of about ½ inch.
Season both sides of the chicken breasts with salt and pepper.
In a shallow dish, mix together the flour, paprika, garlic powder, onion powder, and cayenne pepper (if using). In another shallow dish, whisk together the eggs and buttermilk. In a third shallow dish, place the panko breadcrumbs.
Working with one chicken at a time, dredge the chicken in the flour mixture, shaking off any excess. Dip the chicken into the egg mixture, allowing any excess to drip off. Finally, press the chicken into the panko breadcrumbs, making sure it is evenly coated on both sides.
In a large, deep skillet or dutch oven, heat about 1 inch of vegetable oil over medium-high heat until it reaches 350°F (175°C). Carefully place the breaded chicken into the hot oil and fry for 4-5 minutes per side, or until the chicken is cooked through and the coating is golden brown and crispy. Use a slotted spoon or tongs to remove the chicken from the oil and transfer to a paper towel-lined plate to drain any excess oil.
Spread a layer of mayonnaise or your favorite sandwich spread on the bottom half of each hamburger bun. Top with lettuce, tomato, onion, and pickles. Place a fried chicken breast on top of the vegetables and finish with the top half of the bun.