Add salmon, breadcrumbs, shallots, garlic, green onion, beaten egg, mayo, lemon juice, parsley, salt, and pepper to a large mixing bowl. Mix until well-combined using a wooden spoon, rubber spatula, or your hands.
Divide mixture into 4 equal portions and form into balls, then patties – about ¾ inch thick. *We used a ½ cup measuring cup full of the mixture for each patty. We prefer to use a silicone egg mold to press the mixture into so that it is evenly shaped and well-packed.
Heat oil in a large skillet over medium-high heat. *For griddle cooking, heat griddle to about 375-400℉. Then add frying oil to the griddle for shallow frying.
Cook patties in a single layer, at least 2 inches apart, for about 4 minutes on each side or until golden brown. *Since the salmon mixture contains eggs, be sure to cook to an internal temperature of 160℉.
Carefully remove salmon patties from the skillet or griddle, and transfer to a paper towel lined plate.
Garnish with green onions and/or fresh parsley, and serve with optional yogurt dill dipping sauce and fresh lemons. *See notes.