Combine 3 tablespoons panko bread crumbs, parsley, mayonnaise, lemon juice, scallion, shallot, Dijon mustard, salt, pepper, and pinch cayenne pepper in bowl.
Working in 3 batches, pulse 1 skinless salmon fillet, cut into 1-inch pieces, in food processor until coarsely chopped into ¼-inch pieces, about 2 pulses, transferring each batch to bowl with panko mixture. Gently mix until uniformly combined.
Place remaining ¾ cup panko bread crumbs in pie plate.
Using ⅓-cup measure, scoop level amount of salmon mixture and transfer to baking sheet; repeat to make 8 cakes. Carefully coat each cake in bread crumbs, gently patting into disk measuring 2¾ inches in diameter and 1 inch high. Return coated cakes to baking sheet.
Heat ½ cup vegetable oil in 12-inch skillet over medium-high heat until shimmering.
Place salmon cakes in skillet and cook without moving until golden brown, about 2 minutes.
Carefully flip cakes and cook until second side is golden brown, 2 to 3 minutes. Transfer cakes to paper towel–lined plate to drain 1 minute. Serve.
