Heat grill or stovetop to medium.
Finely chop the salmon. Place half in a large resealable plastic bag and use a flat-surfaced meat pounder or a rolling pin to mash it.
Place all the salmon in a large bowl. Combine with egg whites, mustard, bread crumbs, dill, and ½ teaspoon salt. Form four ¾-inch-thick patties.
Place the patties on grill. Cook 4 to 5 minutes per side. (On a stovetop, heat the oil in a grill pan over medium heat and cook patties 5 minutes per side.)
Serve the patties on the brioche buns, with the toppings of your choice.