Make the salmon mixture. Add the salmon to the base of a food processor and pulse 15 times until broken down into ⅛-inch pieces. Transfer ¾ of the shredded salmon to a large bowl and pulse the remaining salmon until a paste forms, about 15 more times. Add the egg and pulse until incorporated. Add the salmon, salmon paste, panko, onion powder, garlic powder, Old Bay seasoning, mustard powder, lemon juice, and chives and mix until combined.

Form the patties.

Line a plate with parchment paper. Divide the salmon mixture into 4 equal portions and form each into patties.

Cook the burgers.

Heat the oil in a large skillet over medium heat. Add the salmon patties and cook until golden brown on the bottom, about 3 minutes. Flip and add a piece of cheese on top of each patty, then cook until golden brown, 3 minutes more.

Make the Old Bay mayonnaise.

Meanwhile make the Old Bay Mayonnaise. In a small bowl, stir together the mayonnaise, chives, Old Bay seasoning, and lemon juice.

Assemble the burgers.

Spread the Old Bay mayonnaise on the cut sides of the buns. Add the salmon burgers, cheese-sides up, to the bottom buns. Top with caramelized onions, avocado, and top buns.