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1
Combine 1 scant cup of warm milk and yeast in a bowl of a stand mixer fitted with a dough hook. Let sit until foamy and bubbly, about 10 minutes. Add sugar, salt, egg, butter, and almost all of the flour, holding back a small amount in case dough gets too dry.
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2
Knead in the mixer until dough is soft and supple. Scrape dough onto your work surface. Grease the bowl with a few drops of oil and place dough back in. Cover bowl and let dough rise in a warm spot until doubled in volume, about 1 hour.
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3
In the meantime, heat olive oil and butter in a pot over medium-high heat. Add onion, ground beef, and garlic. Season with kosher salt and black pepper. Cook beef, breaking apart with a wooden spoon, until no longer pink, 5 to 7 minutes. Continue cooking, stirring occasionally, until evenly browned, about 2 minutes more. Remove from heat.
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4
Stir dill into the beef mixture. Add chicken broth and stir, scraping up the browned bits from the bottom of the pan. Let mixture cool for 10 minutes. Stir in Cheddar cheese and Parmesan cheese. Let filling cool completely.
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5
Transfer dough to a work surface. Press out air bubbles. Pinch off a piece of dough and form into a ball; press into a disc. Dust with a minimal amount of flour and roll into a circle about 1/8-inch thick and 5 to 6 inches in diameter. Dip your finger in water and dampen the edges of the circle.
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6
Place dough circle in 1 hand and add a few tablespoons of the beef filling. Pinch edges together to seal. Place pirozhki on the table seam-side up and pinch off any excess dough; too much dough will make it hard to fry. Moisten the center of the seam with water. Fold the 2 ends inward and flip pirozhki over, seam-side down; press down lightly.
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7
Form the remaining pirozhki and let them rest until dough rises slightly, 15 to 20 minutes.
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8
Heat oil in a deep-fryer or large saucepan to 375 degrees F (190 degrees C). Fry pirozhki in batches, seam-side down, until golden brown, about 90 seconds. Flip and fry until browned on the other side, about 90 seconds more. Drain on paper towels and cool for a few minutes.
