Prepare salmon: Drain canned salmon or flake cooked salmon with a fork; remove any large bones and skin. Pat dry if moist.
Sauté aromatics (recommended): In a small skillet, heat 1 tsp oil and gently sauté diced onion (and celery if using) 2–3 minutes until softened; cool slightly.
Mix wet ingredients: In a medium bowl whisk the beaten egg, Dijon mustard, mayonnaise (if using), Worcestershire, lemon juice, salt, pepper, and cayenne.
Combine: Add flaked salmon, cooled aromatics, parsley, and 3/4 cup breadcrumbs to the wet mix. Fold gently until evenly combined—stop when mixture holds together when pressed. If too loose add 1 tbsp breadcrumbs at a time; if too dry add 1 tsp mayo or lemon.
Shape patties: Portion 1/3–1/2 cup per patty and form 8–10 patties about 3–3.5 inches wide and 1/2–3/4 inch thick. Place on a plate and chill 10–15 minutes to set.
(Optional) Dredge: For extra crunch, press each chilled patty into extra panko/breadcrumbs to coat.
Heat pan: Over medium heat heat 3 tbsp neutral oil + 1 tbsp butter (optional) in a skillet until shimmering.
Fry: Add patties without crowding. Fry 3–4 minutes per side until deep golden and crisp. Flip gently—try to flip only once. Adjust heat so exterior browns without burning. Work in batches as needed.
Drain & rest: Transfer cooked patties to a rack or paper-towel-lined plate for 1–2 minutes to drain and set.
(Optional) Make remoulade: Whisk remoulade ingredients together and chill until serving.
Serve: Plate with lemon wedges, a dollop of remoulade or tartar sauce, and your choice of sides (slaw, greens, potatoes, or a toasted bun for sandwiches).