Prepare the filling: In a large sauté pan or pot, melt the butter over medium-low heat and add the sliced onion. Sauté until translucent and slightly softened, about 10 minutes.

Add the shredded carrots, shredded cabbage, salt, and pepper, and continue cooking, stirring occasionally, until the vegetables begin to soften and caramelize, about 10-12 minutes.

Stir in the tomato paste and ¼ cup water. Cover and allow the mixture to cook down and braise until the cabbage is deeply caramelized and soft but not mushy, about 7-10 minutes. Uncover the pan to stir occasionally. 

Taste and adjust seasoning. The mixture should be quite peppery. Transfer the filling to a bowl and let it cool completely before using.

Prepare the dough: Gently heat the milk in a microwave or a small pot over medium heat until warm to the touch (105°F).

In the bowl of a stand mixer, add the yeast and sugar. Pour the warm milk over the yeast mixture, whisk just to combine. Allow it to bloom for about 10-15 minutes until the mixture becomes slightly frothy.

Once the yeast has bloomed, attach the dough hook, and add the egg, salt, flour, and melted butter. Mix on low speed for 2-3 minutes, until a shaggy dough forms, then increase to high speed and knead for 7-10 minutes, until smooth and elastic.

Place the dough in a lightly greased bowl, cover with plastic wrap, and allow it to rise for about 30 minutes, until slightly puffed.

Turn the dough out onto a floured surface and punch out the air. Divide the dough into 6 equal portions, about 120 grams each. Form them into balls, cover with plastic wrap and allow them to rest for 5-10 minutes.

Fill a medium heavy-bottom pot with oil halfway (about 3-inches deep) and begin heating it over medium heat until shimmering but not smoking, 350°F.

Meanwhile, on a lightly floured surface, roll each ball into a circle, about 8 inches wide and ¼-inch thick. Place a hefty ⅓ cup of the cabbage filling in the center of each dough circle. Seal tightly by pinching the edges together into an oval shape.

To check if the oil is hot enough, insert the end of a wooden spoon into the oil — if it bubbles gently around the wood, the oil is ready. If it bubbles vigorously, it's too hot. Working in batches, carefully place the prepared piroshki into the hot oil seam-side down and fry until golden brown, flipping halfway. About 3-5 minutes per side. Be careful not to overcrowd the pot. 

Remove the piroshki from the oil using a slotted spoon and let drain on a paper towel lined plate or wire rack. Let piroshki cool for a few minutes and serve.