Stir salmon, eggs, lemon juice, capers, salt, black pepper, and cayenne pepper together in a bowl until well-combined.
Crush saltine crackers with your hands into the salmon mixture and mix well. Wrap the bowl with plastic wrap and refrigerate, 30 minutes to overnight.
Dust a plate with half the bread crumbs. Divide salmon mixture into 4 portions and shape into patties; place onto prepared plate and sprinkle remaining bread crumbs atop the salmon patties.
Melt butter and oil in a large skillet over medium heat. Cook patties in hot oil until cooked and heated through, about 5 minutes per side.
