In a medium bowl, gently combine the flaked salmon, chilled mashed potatoes, the small beaten egg (for the mixture), finely chopped fresh parsley, lemon zest, dried oregano, salt, and freshly ground black pepper.
Mix with a fork until all ingredients are just incorporated. Be careful not to overmix; the salmon should be broken into smaller pieces but retain some texture. If the mixture feels too soft to easily shape, cover the bowl and refrigerate for at least 30 minutes to an hour, which will help it firm up.
Set up your coating station: In a shallow dish or bowl, whisk the other small egg with the 2 teaspoons of water. In a separate shallow dish, spread out the plain breadcrumbs and, if desired, season them with a little salt and pepper.
Divide and shape the salmon-potato mixture into 6 medium-sized patties (this will provide 2 patties per serving). Handle the mixture gently.
Carefully dip each patty into the egg wash, ensuring it's fully moistened on all sides. Allow any excess egg to drip off.
Immediately transfer the egg-washed patty to the dish of breadcrumbs. Dredge the patty, pressing lightly so the crumbs adhere evenly to all surfaces, creating a complete coating.
Place the coated patties on a clean baking sheet or large plate as you finish them.
Heat the olive oil in a large non-stick skillet or frying pan over medium heat. The oil is ready when it shimmers slightly.
Carefully place the salmon patties into the hot skillet, ensuring not to overcrowd the pan (cook in two batches if necessary to maintain an even cooking temperature and make flipping easier).
Pan-fry the patties for approximately 3-5 minutes on the first side, or until they are a rich golden brown and crispy.
Gently flip the patties and cook for an additional 3-5 minutes on the other side, until equally golden brown and heated through. If the pan seems too dry during cooking, you can add a little more olive oil.
Once cooked, transfer the golden patties to a plate lined with paper towels to absorb any excess frying oil.
Serve warm, perhaps with a wedge of lemon, a dollop of yogurt-dill sauce, or a fresh side salad.