Preheat the oven to 375 F. Gently heat sour cream in a small casserole and keep at a simmer.
Make roux by melting 1 tbsp butter over medium heat and sifting 1 tbsp flour over it. Cook, stirring constantly, until the mixture is a deep-golden color. Whisk the roux into sour cream, season with salt, and simmer for 1 minute. Cover the sauce and set aside.
Melt 2 tbsp butter in a skillet. Season the filets with salt and pepper. Roll in flour.
Fry the fish pieces in butter until golden-brown on both sides. Drain on paper towels.
Saute mushrooms in 1 tbsp butter until they have reabsorbed their liquid, about 2 minutes.
Separately, skillet-fry potatoes in vegetable oil until slightly crispy on the outside, but slightly undercooked on the interior. Drain on paper towels.
Arrange the fish pieces on the bottom of a braiser. Top each piece with egg rounds and mushrooms. Arrange potatoes between and around the fish pieces.
Pour the sour cream sauce over the fish and potatoes.
Bake, uncovered, for 30 minutes. Sprinkle with dill and serve.
