Prepare the Fish Cakes: Poach the white fish in simmering water for 5 minutes, then drain and flake with a fork.
Mix Ingredients: In a bowl, combine flaked fish, mashed potato, shallot, dill, parsley, mustard, beaten egg, breadcrumbs, salt, and pepper. Mix until just combined.
Form Patties: Shape the mixture into 4 small fish cakes (or 2 larger ones) and chill for 10 minutes to help them hold their shape.
Cook the Fish Cakes: Heat butter and oil in a skillet over medium heat. Fry fish cakes for 3-4 minutes per side until golden brown and crispy.
Make the Sauce: In a small bowl, mix Greek yogurt, grated cucumber, lemon juice, dill, salt, and pepper. Let sit for flavors to meld.
Prepare the Salad: Toss mixed greens with olive oil, apple cider vinegar, salt, and pepper in a bowl.
Serve: Plate the fish cakes with a dollop of cucumber-yogurt sauce and a side of mixed greens.