1. Pass salmon filet through a meat grinder fitted with a large disc or cut into 1-inch pieces and place in the bowl of a food processor; pulse until ground, 1 to 2 minutes.

2. Transfer ground salmon to a large bowl. Add scallions, lemon and lime zests; mix until well combined. Divide mixture evenly into six portions. Using a 3-inch ring mold, evenly pack one portion of the salmon mixture into ring mold to create a 3/4-inch thick patty. Repeat process with remaining salmon. Place patties on a parchment paper-lined baking sheet and transfer to refrigerator until chilled, about 15 minutes.

3. Preheat a griddle over medium-high heat. Remove salmon patties from refrigerator and brush evenly with butter; season with salt and pepper. Lightly brush griddle with oil and place patties on griddle. Cook, turning once, until medium-rare, 4 to 5 minutes. Remove patties from griddle. Brush cut sides of buns with remaining butter and place, cut side-down, on griddle until toasted, about 1 minute.

4. Place arugula in a small bowl and drizzle with olive oil and lemon juice; toss to combine.

5. Spread 1 tablespoon tartar sauce over the top half of each hamburger bun. Top tarter sauce with 1 tablespoon pickled onions, baby arugula, and 3 slices of avocado each. Place the salmon patties on the bottom half of each bun and sandwich with top half; serve immediately.