While you're organizing your ingredients, preheat a skillet over medium-high heat and lightly toast the buns until they're golden and crispy—this prevents them from getting soggy and adds a nice textural contrast. Set the toasted buns aside on a clean surface. Thinly slice the large onion into 1/4-inch rings and have all your seasonings measured and ready. I like to toast the buns early so they're ready whenever the burgers are done cooking.
In a large bowl, combine the ground chicken, cheddar cheese, 1/4 cup bbq sauce, paprika, onion powder, garlic powder, salt, pepper, and Worcestershire sauce. Mix gently with your hands until just combined—avoid overworking the mixture, as this can make the burgers tough and dense. Divide the mixture into 4 equal portions and shape each into a patty about 3/4-inch thick, making a slight indent in the center with your thumb to prevent them from puffing up during cooking.
While the patties rest for a few minutes, heat 1 tablespoon of olive oil in a separate skillet over medium-high heat. Add the sliced onions, sprinkle with 1/4 teaspoon salt and brown sugar, and cook for about 10 minutes, stirring occasionally, until they're deeply golden and caramelized. The brown sugar enhances their natural sweetness and adds complexity to the burgers.
Heat a skillet over medium-high heat until just smoking. Place the chicken patties in the hot pan and cook for 8 minutes on the first side without moving them—this develops a nice crust and lets the cheese melt slightly. Flip the patties carefully and cook for another 8 minutes on the second side until cooked through. In the last minute of cooking, top each patty with a slice of gouda cheese and let it melt.
Place each cooked burger patty (with melted gouda) on a toasted bun from Step 1. Top with a generous handful of fresh romaine lettuce, a portion of the caramelized onions from Step 3, and a drizzle of additional bbq sauce. Serve immediately while the burgers are still warm and the buns are crispy.