Preheat the oven to the "keep warm" setting (170℉).
Drain the salmon cans thoroughly. Place the salmon in a large bowl. Flake it well with a fork into tiny pieces.
Add the eggs, mayonnaise, Dijon mustard, garlic, kosher salt, black pepper, and thyme. Mix well.
Mix in the chopped parsley.
Heat the olive oil over medium heat in a large 12-inch nonstick skillet for about 3 minutes.
Using a 4-tablespoon ice cream scoop, transfer four mounds of the mixture to the skillet. Pack the mixture into the scoop, then release it into the skillet and gently flatten.
Cook the salmon patties until well-browned and crispy, about 3 minutes on each side. Place the cooked patties in the oven to keep them warm while you cook the second batch.
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Notes
To make sure the cakes don’t fall apart, drain the salmon thoroughly, flake it well, use large eggs, and mix the ingredients thoroughly.
Different canned salmon products can have different textures. It's important to drain the salmon so that it's visibly dry - place it on paper towels to drain if needed. If, despite your efforts, you mix the ingredients and the mixture seems very liquid, add 2-4 tablespoons of breadcrumbs (regular or gluten-free) or (for a low-carb, grain-free version) almond flour.
About half the olive oil will remain in the skillet. The nutrition info reflects that. But you need all of it to fry the patties properly.
To bake these patties, line a rimmed baking sheet with parchment paper. Measure ¼ cup per patty with a scoop, packing the mixture in. Place on the parchment and gently flatten. Spray the patties with olive oil and bake them for 10 minutes in a preheated 400°F oven. Flip, spray with more oil, and bake for 10 more minutes. If you’d like to brown them, replace the parchment with foil and broil them in the middle of the oven for 2 minutes.
You can keep the leftovers in an airtight container in the fridge for 3-4 days. Reheat them gently in the microwave at 50% power, or enjoy them cold straight out of the fridge. You can also freeze these patties in a single layer in freezer bags for up to three months.