Heat butter in a large pot over medium heat. Add onions and cook until softened, about 5 minutes. Add mushrooms and sauté for an additional 5 minutes.
Stir in vegetable broth, soy sauce, dill, and paprika. Reduce heat to low, cover, and simmer for 15 minutes.
In a separate bowl, whisk together milk and flour until smooth. Stir this mixture into the soup. Cover and simmer for another 15 minutes, stirring occasionally.