Place the cubed potatoes into a pot of salted water and cook until tender, around 15 to 20 minutes. Drain and allow them to cool slightly.
Mash the potatoes thoroughly and wait until they are cool before adding the egg and salt, then mix together well.
Slowly incorporate the flour into the potato mixture until a pliable dough forms; add more flour if it feels sticky but avoid overmixing.
Flour your surface lightly and split the dough into four sections. Roll each into a rope about 2 cm thick.
Cut the dough ropes into pieces approximately 2.5 cm long.
Bring a large pot of salted water to a boil and carefully drop in the kopytka in batches.
Allow them to float up and keep them cooking for two additional minutes before removing with a slotted spoon; set aside.
Warm olive oil and butter in a skillet over medium heat. Sauté the onion until softened, about 3 minutes.
Add the garlic and cook for 30 seconds more.
Introduce the mushrooms to the pan and cook until golden and moisture evaporates, about 5 to 7 minutes.
Pour in the vegetable broth and add thyme, letting it simmer for 5 minutes.
Stir in the cream or sour cream, season with salt and pepper, then simmer until the sauce thickens slightly, around 2 minutes.
Add the cooked kopytka to the mushroom sauce and gently toss to coat everything well.
Heat through the dumplings in the sauce for 2 to 3 minutes before serving.
Garnish with chopped fresh parsley and sprinkle grated Parmesan cheese if desired.
Serve the dish warm to enjoy the rich flavors.
