Step 1. Cut salmon into 2″ pieces. Transfer one-third of salmon (about 8 oz.) to a food processor and process, scraping down sides, until mixture is very smooth and paste-like. Add remaining salmon and pulse 4–5 times until pieces are no bigger than ¼” (be careful not to make it too smooth). Transfer to a large bowl.
Step 2. Mix in scallion greens, ginger, garlic, 2 Tbsp. mayonnaise, and 1 tsp. salt; toss to combine. Form mixture into 4 patties about ¾” thick. Transfer to a parchment-lined rimmed baking sheet and cover with plastic wrap. Chill at least 1 hour or up to 3 (you’ll want to chill the patties so that they hold their shape before getting cooked).
Step 3. Meanwhile, mix sesame oil, 1 tsp. vinegar, remaining ⅔ cup mayonnaise, and a pinch of salt in a small bowl; set aside until ready to use.
Step 4. Toss cucumbers with a pinch of salt in another small bowl. Massage with your hands for a few minutes, squeezing lightly to expel water; discard cucumber liquid. Add chile, sugar, and 2 tsp. vinegar to bowl; toss to coat. Chill until ready to assemble burgers.
Step 5. Heat oil in a large nonstick skillet over medium-high until oil begins to shimmer. Remove salmon patties from fridge right before cooking and sprinkle with flour just to coat the outside (you won’t need all of it). The patties will be a little loose but you can always pat them back together with your hands before they hit the pan. Working in batches and adding more oil in between batches if needed, cook patties until golden brown, about 3–4 minutes on each side (you don’t want to overcook).
Step 6. Toss herbs, watercress, sesame seeds, if using, scallion whites, remaining 1 tsp. vinegar, and a pinch of salt in a medium bowl. Build burgers with buns, patties, reserved special sauce, herb mixture, and pickles.