Cut half of the salmon into large chunks, and place into the bowl of a food processor. Add the lemon juice, mayonnaise, and mustard. Process into a paste.
1½ pounds skinless, boneless salmon, 1 tablespoon lemon juice, 1 tablespoon mayonnaise, 2 teaspoons Dijon mustard
Dice the remaining salmon and place in a large mixing bowl. Add the green onions, ¼ cup of breadcrumbs, salt, pepper, and salmon paste from the food processor. Mix gently to combine.
2 green onions, ¾ teaspoon kosher salt, ½ teaspoon ground black pepper, 1¼ cups Panko breadcrumbs
Divide the mixture into four equal mounds and shape into patties. Place the patties on a plate, cover, and chill for 30 minutes (up to 2 hours).
Spread the remaining 1 cup of breadcrumbs onto a plate. Press both sides of each salmon patty in the breadcrumbs.
Heat the butter in a large skillet over medium-high heat. Add the patties and cook for 3-4 minutes until browned. Flip and cook for an additional 3-4 minutes, being sure not to overcook. Transfer to a paper towel-lined plate.
2 tablespoons unsalted butter
Serve on buns or a bed of greens with lemon wedges, tartar sauce, and hot sauce.
Lemon wedges, tartar sauce, and hot sauce