Cut the salmon into large chunks. Add about one-third of the salmon and the mustard to a food processor. Process until the mixture becomes a smooth paste, scraping down the sides as needed.

Add the shallots and the remaining salmon to the processor. Pulse on and off until everything is well combined and the fish is coarsely chopped — no pieces larger than ¼ inch. Be careful not to over-process; you want texture, not a puree.

Transfer the mixture to a bowl. Stir in the bread crumbs, capers, salt, and pepper by hand. Shape into four even patties. (At this point, you can cover and refrigerate them for a few hours if needed.)

Heat butter or oil in a 12-inch nonstick skillet over medium-high heat. When the butter foams or the oil is hot, cook the burgers for 2 to 3 minutes per side, turning once.

Grill over medium-high heat for about 4 minutes on the first side to firm up, then flip and cook 1 to 2 minutes on the other side.

Check for doneness by making a small cut — the center should be just cooked through. Avoid overcooking.