Combine salmon, eggs, and onion in a bowl. Season with cayenne and salt; mix well. Place fish fry in another bowl.
Heat oil in a skillet over medium-high heat.
While the oil heats, place a large spoonful of salmon in the palm of your hand and form an oval croquette patty. Place croquettes in the fish fry and flip over to coat. Repeat to form remaining croquettes.
Working in batches, place croquettes in the hot oil and cook until edges have browned, about 5 minutes. Flip and cook the other side, about 5 minutes. Remove to a paper towel-lined platter and repeat to cook remaining batches.