In a large bowl, whisk together flour and salt. Make a well in the center and add egg, warm water, and oil. Stir with a fork until shaggy, then turn onto a floured surface and knead for 5-7 minutes until smooth and elastic. The dough should feel soft but not sticky - add flour 1 tablespoon at a time if needed. Cover with a damp towel and rest for 20 minutes.

Press farmer's cheese through a fine-mesh sieve or ricer into a bowl to remove any lumps. Stir in egg yolk, sugar, and salt until well combined. The filling should hold together when pinched but not be wet - if too moist, blot with paper towels.

Divide dough in half. Roll one portion on a floured surface to 1/8-inch thickness - thin enough to see your hand through it slightly. Cut into 3-inch circles using a cutter or glass. Gather scraps, reroll once, and cut again. Repeat with remaining dough.

Place 1 teaspoon of filling in the center of each circle. Fold dough over to form a half-moon, pressing edges firmly to seal. Pick up each dumpling and pinch the curved edge into small pleats or simply press with fork tines to ensure a tight seal - this prevents bursting during cooking. Place on a floured sheet pan, not touching.

Bring a large pot of salted water to a gentle boil. Working in batches of 8-10, slide dumplings into the water. Stir once gently to prevent sticking. Cook until they float to the surface, about 3-4 minutes, then continue cooking 1 minute more. Remove with a slotted spoon to a plate.

Heat 2 tablespoons butter in a large skillet over medium heat. Add half the boiled dumplings in a single layer and cook undisturbed until golden and crisp on the bottom, 2-3 minutes. Transfer to a serving dish, add remaining butter, and repeat with remaining dumplings.

Arrange dumplings on a warm platter, crispy sides up. Top with a dollop of sour cream and sprinkle with fresh dill. Serve immediately with extra sour cream on the side.