Cut salmon into smaller chunks and put it into a container of a food processor. Turn it on and let it run until the mixture becomes pasty, stopping to scrape down the sides if needed.
Once done, add mayonnaise, 4 tablespoons of breadcrumbs, salt, pepper, lemon juice, mustard, cayenne, onion to the food processor. (If using parsley, add it in there now). Pulse a few times until all the ingredients are combined.
Add the remaining breadcrumbs to a medium bowl. Divide ground salmon mixture into 6 equal parts or 4 large and gently form into balls. To make it easier (they can be slightly sticky but should shape nicely) have a small bowl with warm water next to you to dip your fingers in it. Once the balls formed, dip each patty into breadcrumbs and flatten slightly.
Warm-up ⅓ cup oil over medium-high heat in a large skillet and once hot fry patties on both sides until crisp and golden brown (about 2-3 minutes per side). You may need to do it in batches. Transfer the patties to a plate lined with paper towels to drain the excess oil.
Squeeze fresh lemon juice on each patty before serving (optional).
Assemble the burgers with favorite lettuce and vegetables, add mayonnaise and a drizzle of hot sauce on both bottom and top buns, serve while warm.
