Clean the fresh mussels.
Add 3 Tbsp water, 3–4 sprigs of lovage, and the sliced garlic to a pot. Add the mussels, cover, and cook over high heat until the shells open.
Transfer the cooked mussels to a tray and let them cool enough to handle. Discard any mussels that remain closed after cooking. Strain the liquid from the pot and reserve it. Remove the mussel meat from the shells and set it aside.
Blanch the tomato, peel it, and chop it finely. Finely chop the onion and carrot.
Heat the olive oil in a soup pot. Add the onion and carrot, cover, and cook until softened.
Pour in all the strained mussel liquid and 5 cups water. Add the bay leaf and bring to a boil. Add the tomato, reduce the heat, and cook for 10 minutes.
Add the rice and cook until tender. Turn off the heat and let the soup cool a little.
Whisk the egg yolk and Bulgarian yogurt together until evenly combined. Add the light cream and whisk again.
Slowly add 3 ladles of hot soup to the yogurt mixture, whisking continuously. Add a fourth ladle if needed. Slowly pour the mixture back into the soup, stirring continuously.
Add the salt, black pepper, cooked mussels, and finely chopped lovage. Taste and adjust the seasoning if needed.