Prepare all ingredients as instructed above.
In a medium mixing bowl, combine the ground beef, salt, black pepper, smoked paprika, Carolina-style tangy BBQ sauce, and Worcestershire sauce. Mix gently until just combined.
Shape the beef mixture into evenly sized burger patties.
Heat a pan or grill over medium-high heat. Cook the burger patties on both sides to your liking.
Place a slice of cheddar cheese on each patty during the final minute of cooking and cover to allow the cheese to melt.
Toast the hamburger buns on the grill or in a pan until golden brown, about 2–3 minutes.
Assemble the burgers by layering the bottom bun with arugula, tomato slices, pickle slices, and the cheeseburger patty. Drizzle with additional BBQ sauce if desired, then top with the remaining bun half.
Combine the sparkling water with the lemon slices to make the lemon sparkling water. Serve over ice.
Serve the burgers with carrot sticks, hummus, and the lemon sparkling water.