In a large mixing bowl, whisk together the all-purpose flour, salt, sugar, and instant yeast. Gradually pour in the warm milk and warm water, followed by the vegetable oil and beaten egg. Mix until the dough begins to form, then transfer it to a floured surface and knead for about 5-7 minutes until smooth and elastic.

Place the kneaded dough in a lightly oiled bowl, covering it with a damp cloth or plastic wrap. Allow it to rise in a warm, draft-free area for about 1 hour or until it doubles in size.

While the dough is rising, prepare the filling by heating butter in a skillet over medium heat. Add the chopped onions and optional mushrooms, sautéing until softened and translucent, about 5 minutes. Stir in the ground beef, season with salt and pepper, and cook until the meat is browned and fully cooked through, which should take about 7-10 minutes.

Once the dough has risen, punch it down gently and roll it out on a floured surface to about 1/4 inch thickness. Cut the dough into 12 equal squares or circles.

Place a generous spoonful of the beef filling in the center of each piece, fold the dough over, and pinch the edges tightly to seal.

Preheat your oven to 375°F (190°C) and line a baking sheet with parchment paper. Arrange the shaped piroshki on the baking sheet, allowing a little space between each. Brush the tops with a beaten egg for a beautiful golden color, then bake for 15-20 minutes or until they are golden brown and puffed.

Once baked, remove the piroshki from the oven and let them cool slightly on a wire rack. Serve warm.