Step 1: Prepare the Kopytka Dough

1. Boil the cubed potatoes in salted water until fork-tender, about 15-20 minutes. Drain well and let them cool slightly.

2. Mash the potatoes until smooth and let them cool completely.

3. Add the egg and salt to the mashed potatoes and mix well.

4. Gradually add the flour, mixing until a soft, non-sticky dough forms. If the dough is too sticky, add a little more flour, but avoid overworking it.

Step 2: Shape and Cook the Kopytka

1. Lightly flour a work surface and divide the dough into 4 portions.

2. Roll each portion into a long rope about 3/4 inch (2 cm) thick.

3. Cut each rope into 1-inch (2.5 cm) pieces.

4. Bring a large pot of salted water to a boil. Drop the kopytka in small batches into the boiling water.

5. Once they float to the surface, let them cook for another 2 minutes, then remove with a slotted spoon and set aside.

Step 3: Prepare the Mushroom Sauce

1. Heat the olive oil and butter in a large skillet over medium heat.

2. Add the chopped onion and sauté until soft, about 3 minutes.

3. Stir in the garlic and cook for another 30 seconds.

4. Add the sliced mushrooms and cook until they release their moisture and become golden brown, about 5-7 minutes.

5. Pour in the vegetable broth and thyme, then simmer for 5 minutes.

6. Stir in the heavy cream or sour cream and season with salt and pepper to taste. Simmer for another 2 minutes until the sauce thickens slightly.

Step 4: Combine and Serve

1. Add the cooked kopytka to the skillet with the mushroom sauce and toss gently to coat.

2. Cook for an additional 2-3 minutes until heated through.

3. Garnish with fresh parsley and grated Parmesan cheese, if desired.

4. Serve warm and enjoy!