Cut the chicken breasts into chunks and place in a food processor. Pulse until coarsely ground but not pureed.
Transfer the ground chicken to a mixing bowl. Add chopped onion, minced garlic, breadcrumbs, egg, salt, black pepper, smoked paprika, and dried oregano. Mix until combined.
Divide the mixture into 4 equal portions and shape each into a patty about 3/4 inch thick.
Heat olive oil in a large skillet or grill pan over medium heat.
Cook the chicken patties for 6 to 7 minutes on each side, or until the internal temperature reaches 165°F (74°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 pan to melt the cheese.
Toast the hamburger buns lightly on the grill or in a toaster.
Assemble the burgers by placing a lettuce leaf and tomato slice on the bottom half of each bun, then add the chicken patty with melted cheese. Top with the other half of the bun.
Serve immediately and enjoy your chicken breast burgers.