Place half of salmon in a food processor and pulse until finely chopped, about 10 short pulses. Transfer to a large bowl and repeat with remaining salmon. Alternatively, chop salmon by hand and transfer to a large bowl. Finely mince half of the onion and add to salmon. Thinly slice remaining half onion and set aside.

Add half of the dill, 1 ½ teaspoons of the honey, and half of the horseradish to the salmon. Season lightly with salt and pepper and knead with hands until homogenous. Form into 4 patties about 4 inches across and ⅓-inch thick. Season exteriors with salt and pepper and set aside.

Combine mayonnaise, dijon mustard, remaining honey, and remaining dill in a small bowl and stir until homogenous.

Heat the oil in a 12-inch skillet over medium heat until shimmering. Add the patties and cook until browned, about 2 minutes. Flip and cook until the burgers are just barely cooked through and still slightly rare in the center, about 2 minutes longer. Transfer to a paper towel-lined tray and allow to rest for 2 minutes.

Divide lettuce and onion between the four bottom buns and place a salmon patty on top. Spoon honey mustard over them, then top with avocado and top bun. Serve immediately.