Using your hands, thoroughly combine salmon, chopped parsley, celery, onion, egg, 1 tablespoon garlic, cayenne, salt and pepper in a large mixing bowl. Mix until it begins to bind.
Rub your hand with a bit of avocado oil to prevent the mixture from sticking. Then scoop, with a 1/4 cup measuring cup, and delicately form into patties.
Preheat a non-stick, cast-iron or high-quality stainless steel skillet over medium heat. Add 2-3 tablespoons of avocado oil and allow oil to come up to temperature before adding patties. Work in batches to avoid overcrowding.
Cook patties 3 minutes; then flip using a slotted fish spatula. Cook an additional 3 minutes for well-done. Cook 2 minutes per side for a medium-rare center.
Meanwhile, using a mandoline or sharp chef’s knife, slice fennel into paper thin strips. Whisk together olive oil, lemon juice and Dijon mustard and season with salt and pepper. Toss dressing with fennel, bibb lettuce and parsley. Remove salmon cakes from pan and serve with salad and aioli, if desired.