Cut the salmon into smaller chunks and put into the bowl of a food processor. Turn on the processor and let it run until the mixture becomes paste-like. Stop and scrape down the sides of the bowl if needed.
Once the salmon is pureed, add the onion, 1/4 cup of breadcrumbs, mayo, lemon juice, Dijon, parsley (if using), salt, pepper, and cayenne to the bowl. Pulse a few times until all the ingredients are combined.
Divide the salmon mixture into 6 equal portions and gently form into balls. (To make it easier have a small bowl with warm water next to you to dip your fingers in it.)
Spread the remaining 3/4 cup of breadcrumbs onto a plate. Then dip each salmon ball into the breadcrumbs and flatten slightly before flipping to coat the other side to form patties.
Place a large skillet over medium-high heat and add in the oil. Once hot, fry the patties for 2-3 minutes on each side until crisp and golden brown. Don't overcrowd the pan - you may need to cook in batches. Transfer the patties to a plate lined with paper towels to drain the excess oil.
Assemble the burgers with some mayo and a drizzle of hot sauce spread on the top and bottom buns. Top the sauce with lettuce, a salmon burger patty, tomato slice, and some sliced red onion. Serve warm.
Store the patties in an airtight container in the fridge for up to 2 days or freeze for up to 3 months.