Rehydrate mushrooms in a small bowl filled with water until they become tender, for 20 to 30 minutes. Save the flavorful soaking liquid for later use.
In a large pot over medium-high heat, melt 1/4 cup of butter until sizzling. Sauté onions, veal, ham, sausage, and rehydrated mushrooms in the fragrant butter until golden. Pour in stock and mushroom water, then bring to a vigorous boil.
Create a fragrant bouquet garni by tightly wrapping bay leaves and peppercorns in cheesecloth. Introduce the bouquet garni, marinated mushrooms, pickles, and capers to the soup. Simmer gently for 10 to 15 minutes to infuse the flavors.
Melt the remaining 1/4 cup butter in a skillet. Add tomatoes and tomato paste; cook for 2 to 3 minutes. Then add flour and cook for another 2 to 3 minutes. Stir in 1 cup of soup, then return the mixture to the soup pot.
Combine olives, dill, pickle juice, garlic, paprika, and marjoram in a pot. Season generously with salt and black pepper. Simmer for 10 to 15 minutes.
Take the pot off the heat and remove the bouquet garni. Taste and adjust seasonings, then serve with a dollop of sour cream and a slice of lemon to garnish.