In a bowl, mix flour, salt, and egg. Add water gradually and knead into a firm, smooth dough. Cover and rest for 30 minutes.
In a separate bowl, combine ground pork, onion, garlic, salt, and black pepper until cohesive and slightly sticky.
Roll the dough thin and cut into small circles. Place a teaspoon of filling in each, fold into half moons, and seal the edges. Bring the corners together to form pelmeni.
Bring a pot of salted water to a gentle boil. Add pelmeni and cook until they float, then continue cooking for 1 to 2 minutes more. Remove and drain.
Heat butter in a skillet over medium heat. Add the boiled pelmeni and pan fry until golden and crisp on both sides.
Meanwhile, toss diced potatoes with olive oil, salt, black pepper, and paprika. Roast at 400°F until golden and tender, about 30 to 35 minutes.
Serve the pelmeni and potatoes together with a dollop of sour cream and a sprinkle of fresh dill.