FOR Salmon Burgers: Cut the salmon into 2-inch chunks and put into the bowl of a food processor fitted with a metal blade. Pulse until the fish is broken down into a chunky puree. It should still have pebble-sized pieces (about ¼-inch) in it. Scrape into a bowl.
CUT the root end from the green onions, slice in half lengthwise, turn them 90° and slice them very thinly. Put this into the bowl with the salmon, ½ cup of the panko bread crumbs, mayonnaise, and pickled ginger liquid. Mix until uniform.
LINE a baking sheet with plastic wrap or parchment paper. Divide the salmon mixture into 8 mounds and form those mounds into patties that are ¾-inch thick. Cover with plastic wrap and refrigerate for 1 to 8 hours before cooking.
POUR the remaining panko bread crumbs onto a plate. Gently lift and place each salmon patty on the bread crumbs, press down lightly to help the panko stick, flip the patties over and repeat.
HEAT a large, heavy-bottomed skillet over medium-high heat. When it is very hot, add about 1 tablespoon of oil to the pan and swirl it to coat.
EASE the panko crusted patties into the pan. A 12-inch pan should hold 4 patties at a time and still give you room to maneuver the spatula when it comes time to flip them. Cook for 5 minutes, or until the first side is deep golden brown. Carefully flip the patties using a wide spatula. Be cautious because the oil will want to splatter. Fry the second side for an additional 3 to 5 minutes, or until golden brown and firm around the outer edges. Transfer to a paper towel-lined plate or platter. Add another tablespoon of oil to the pan and repeat until you have fried all of the salmon burgers.
FOR Wasabi Mayonnaise: Put the mayonnaise in a small bowl and use a fork or small whisk to stir in the wasabi paste, starting with 1 tablespoon. Taste the mayonnaise and adjust with more wasabi if you want a stronger flavor.
TO Serve the Salmon Burgers: If you’re watching how much bread you eat, serve the Salmon Burgers on a tender lettuce leaf topped with a dollop of the Wasabi Mayonnaise and any other optional toppings you’d like. If you want to eat it burger-style, put a patty on top of a toasted whole wheat bun, spread some wasabi mayonnaise over the burger and top as desired.