For the Burgers: Cut the salmon into 2-inch chunks and throw it into the bowl of a food processor. Pulse until it has broken down a little bit but is not smooth. It should still have about ¼-inch pieces in it. Scrape it into a bowl.
Cut off the white part of the green onions and save for another use. Slice the green parts in half lengthwise, then chop them very thinly. Put this into the bowl with the salmon, ½ cup of the panko bread crumbs, 1/4 cup mayonnaise, and pickled liquid. Mix until combined, but don't overdo it.
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. Try to make the edges just as thick as the centers. Cover with plastic wrap and chill for 1 hour (up to 8) before cooking.
Pour the remaining panko bread crumbs onto a plate. Using a large spatula, 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 hot, add about 1 tablespoon of oil to the pan and swirl it to coat.
Gently slide the salmon burgers into the pan. A 12-inch pan should hold 4 patties at a time. Cook for 3-5 minutes, or until the first side is golden brown. Carefully flip the patties using a wide spatula. Use a splatter shield so that your arms don't suffer Death By Hot Oil.
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 platter. Add another tablespoon of oil to the pan and repeat until all of the salmon burgers are done.
For the Wasabi Mayonnaise: put the mayonnaise in a small bowl and use a fork or spoon to stir in the wasabi sauce, starting with 1 tablespoon. Taste the mayonnaise and add more mayonnaise or wasabi to taste.
To serve: top toasted buns with salmon burgers, Wasabi Mayonnaise, and any other toppings that sound good to you. These work great as lettuce wraps. Enjoy!