In a large bowl or the bowl of a stand mixer, combine the flour, instant yeast, sugar, and salt. In a separate bowl or jug, whisk together the warm milk, egg, melted butter, and sour cream until smooth. Add the wet ingredients to the flour mixture and mix until a shaggy dough forms. Knead by hand for 10–12 minutes or in a stand mixer for 8–10 minutes until the dough is smooth, elastic, and slightly tacky but not wet. Shape into a ball and place in a lightly oiled bowl. Cover and let rise for 1 to 1¼ hours, until doubled. The dough should feel soft and supple, not stiff. A slightly tacky dough gives the soft bun texture you want. That sour cream folded into the dough adds real fat and moisture beyond what milk and butter alone would provide, contributing directly to the finished bun's characteristic soft, tender crumb.

While the dough rises, heat the oil or butter in a skillet over medium heat. Add the onion and cook for 6–8 minutes until softened and lightly golden. Add the garlic and cook for 30 seconds. Add the ground beef and cook, breaking it up well, until browned and fully cooked through. Season with the salt, black pepper, and paprika. If the filling looks dry, add 2–3 tablespoons of beef stock or water and cook briefly until absorbed. If using herbs, stir them in at the end. Transfer the filling to a plate or tray and cool completely before using. The filling must be savoury and moist, but not wet or greasy. A hot or wet filling will soften the dough and make sealing difficult. That specific, checkable texture — moist but genuinely not wet — protects the dough's own ability to seal cleanly around the filling once shaping begins.

Punch down the risen dough and turn it out onto a lightly floured surface. Divide into 12 equal pieces, about 70–75g each. Shape each piece into a ball. Cover loosely and let rest for 10 minutes. This short rest relaxes the gluten and makes shaping easier. That relaxation matters directly at the next stage — dough that's just been divided and shaped into balls is genuinely tighter and more resistant to being flattened and filled cleanly.

Working one piece at a time, flatten or roll each dough ball into an oval or round about 10–12cm across. Place about 2 tablespoons of cooled beef filling in the center. Do not overfill. Bring the edges up over the filling and pinch firmly to seal. Shape gently into a smooth oval bun and place seam-side down on a parchment-lined baking tray. Repeat with the remaining dough and filling, leaving space between each bun. Pinch the seam thoroughly. If the seam is weak, the piroshki may split during proofing or baking. That thorough, deliberate pinching along the entire seam is what protects each piroshki from splitting open during the two remaining stages — the final proof and the oven's own heat — where any weak point would be exposed under continued pressure.

Cover the shaped piroshki loosely and let proof for 30–40 minutes, until visibly puffy. They should look slightly expanded and airy, but not overinflated. Meanwhile, preheat the oven to 190°C. That specific, checkable target — puffy but not overinflated — protects the piroshki's own shape and seal, since dough given too much time to rise before baking would put real additional strain on a seam that's already been asked to hold the filling securely.

Whisk together the egg and milk for the egg wash. Brush the tops of the proofed piroshki evenly. Bake for 18–22 minutes, until glossy, deep golden-brown, and fully baked. If you want a softer finish, brush lightly with a little melted butter as soon as they come out of the oven. That even, thorough egg wash is what produces the piroshki's characteristic glossy, deep golden-brown finish rather than a duller, more matte surface.

Let the piroshki cool for 10 minutes before serving. Serve warm. The bread should be soft and fluffy, and the filling should be juicy and well seasoned. That brief cooling period lets the filling's own residual heat settle slightly, making the piroshki easier to handle and eat without the hot filling spilling out the moment they're bitten into.