In a food processor, combine the kimchi and scallions, then process until finely chopped, 30 to 60 seconds. Transfer ½ cup of the kimchi mixture to a medium bowl. Transfer the remaining mixture to a small bowl, then stir in the mayonnaise, kimchi juice and ¼ teaspoon each salt and pepper; reserve the food processor bowl and blade (no need to clean them). Cover the mayonnaise mixture and refrigerate until ready to serve.

To the food processor, add half of the salmon. Process until smooth, about 1 minute, scraping the bowl as needed. Transfer to the medium bowl with the reserved kimchi mixture and stir to combine. To the processor, add the remaining salmon and pulse until chopped, 3 to 4 pulses. Stir the chopped salmon into the kimchi-salmon mixture. Add the gochujang, sesame oil and ½ teaspoon each salt and pepper; stir until well combined.

With wet hands, divide the kimchi-salmon mixture into 4 even portions. Form each portion into a 3-inch patty and place on a large plate. Refrigerate, uncovered, to firm up the patties, about 15 minutes.

If using panko, put the panko in a pie plate or other wide, shallow dish. One at a time, coat the patties with the panko, gently pressing to adhere and reshaping the patties if needed. Return them to the plate.

In a 12-inch nonstick skillet over medium, heat the grapeseed oil until shimmering. Add the patties and cook until browned on the bottoms, 3 to 4 minutes. Using a wide spatula, flip and cook, adjusting the heat as needed, until browned on the second sides, about another 3 minutes. Transfer to a paper towel–lined plate. Serve on the buns with the kimchi mayonnaise.