Cut three-quarters of the salmon (just eyeball it) into ¼-inch pieces. Put in a large bowl. Cut the rest of the salmon into chunks; transfer the chunks to a food processor along with the mustard, egg, lemon zest, lemon juice and cayenne. Pulse to make a paste.

Add the salmon paste mixture to the bowl with the diced salmon. Add the minced shallot, panko, parsley, salt, and black pepper to taste. Gently mix until just combined. The mixture will seem like a sloppy mess.

Line a platter or a baking sheet with parchment paper or wax paper. Divide the salmon mixture into 4 mounds on the parchment paper (if you started with 1 1/4 pounds of salmon, you can probably squeeze out 5 patties.) Form into 4-inch-wide, ¾-inch-thick patties. Cover and refrigerate at least 30 minutes, up to 12 hours ahead. This step helps bind the burgers.

Heat the oil in a large non-stick or cast-iron skillet over medium-high heat or 375 degrees on a griddle. Add the patties (in batches if necessary) and cook until browned on the bottom, about 4-5 minutes, adjusting the heat if necessary. Turn and cook until the other side is browned and the patties feel springy in the center, 4-5 minutes. Serve immediately with desired accompaniments.