Dill citrus aioli: Combine all ingredients in a small bowl. Season with salt and pepper to taste, then set aside.
Burgers: Cut salmon into 2-inch (5-cm) pieces. Transfer 1⁄3 of salmon to a food processor and process, scraping down sides, until mixture is very smooth and paste-like. Add remaining salmon and pulse 4–5 times until pieces are no bigger than 1⁄4 inches (6 mm) (be careful not to make it too smooth). Transfer to a large bowl.
Gently fold in citrus zests, juice, chives, and salt. Shape into 4 burger patties.
Heat 1 ½ teaspoons of oil in a cast-iron skillet over medium heat. Once oil is hot, add salmon burgers to pan. Sear for 3–4 minutes on each side until cooked through.
Assembly: Spread about 1 tablespoon of dill citrus aioli on the bottom of 4 buns. Place one burger on each bun, then top with tomato, avocado, and butter lettuce.
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