In a large bowl, combine ground chicken, finely chopped onion, minced garlic, breadcrumbs, egg, salt, black pepper, smoked paprika, and cayenne pepper. Mix gently until just combined.
Divide the mixture into 4 equal portions and shape each into a round patty about 3/4 inch thick.
Heat olive oil in a large skillet over medium heat.
Cook the chicken patties for 5-6 minutes on each side, or until the internal temperature reaches 165°F (75°C) and the patties are golden brown.
During the last minute of cooking, place a slice of cheddar cheese on each patty and cover the skillet to melt the cheese.
Toast the hamburger buns lightly in a separate pan or toaster.
Spread 1 tablespoon of mayonnaise on the bottom half of each bun and 1 teaspoon of mustard on the top half.
Place a lettuce leaf on the bottom bun, followed by the chicken patty with melted cheese, then a slice of tomato.
Cover with the top bun and serve immediately.