Step 1Drain the salmon, then flake it into a bowl. Add the egg; mix well. Set aside.
Step 2Melt the butter in a skillet over medium heat. Add the onion, bell pepper, garlic and celery and cook until soft, 6 to 8 minutes. Add the vegetables to the salmon and egg mixture. Season with salt and pepper and mix thoroughly. Refrigerate, covered, until the mixture is firm, at least 2 hours.
Step 3To make patties, shape about 1/2 cup of the mixture into ovals. Coat the patties thoroughly on both sides with the cornmeal.
Step 4Heat the oil in a frying pan over medium heat. Fry the cakes, a few at a time, until golden brown on one side, about 5 minutes. Carefully turn them and brown the other side, 4 to 5 minutes. Remove the patties from the pan and place on paper towels to drain.