Sift flour into a bowl. Add finely chopped dill and a pinch of salt, mixing to distribute evenly. Pour in water and knead into a firm, dense dough. Wrap tightly in plastic wrap and refrigerate for 20 minutes to allow the gluten to develop.

Cut the carrot in half. Grate one half finely and reserve the other half for later. Finely chop the onion.

Heat vegetable oil and butter in a pan over medium heat. Add the chopped onion and grated carrot. Sauté gently for 3-4 minutes until softened and fragrant but not browned. Remove from heat and set aside.

Season the ground beef with salt and black pepper. Mix thoroughly to distribute the seasonings evenly.

Remove the dough from the refrigerator and roll it out very thin, about 2-3mm thick, into a rectangular shape. Spread the seasoned ground beef evenly over the entire surface of the dough using your hands or a spatula. Cut the dough lengthwise into two equal strips.

Starting from one long edge, tightly roll each strip into a log, rolling from the center toward the free edge to seal properly. Cut each log crosswise into pieces approximately 2cm wide, creating spiral-shaped dumplings.

Bring 3 liters of water to a boil in a large pot. Cut the remaining carrot half into small cubes and add to the pot. Add the potato cubes along with the bay leaf to the boiling water.

Let the vegetables boil for 5 minutes, then season the broth with salt to taste.

Carefully lower the strulli dumplings into the boiling soup. After the soup returns to a boil, add the sautéed onion and carrot mixture.

Let the soup simmer for 7 minutes until the dumplings are cooked through and the beef is fully cooked.

Turn off the heat and stir in freshly chopped parsley and dill. Ladle the hot soup into bowls, ensuring each serving has several strulli dumplings and plenty of vegetables.