Melt the butter in a large pot over medium heat. Add the onions and cook until softened, about 5-7 minutes.
Stir in the mushrooms and cook until they release their moisture and begin to brown, about 10 minutes.
Add the paprika and dried dill, stirring to coat the mushrooms.
Sprinkle the flour over the mushroom mixture and stir to combine. Cook for 1-2 minutes to remove the raw flour taste.
Gradually whisk in the broth, milk, and soy sauce. Bring to a simmer, stirring frequently until the soup thickens, about 10 minutes.
Remove the pot from heat and stir in the sour cream, lemon juice, and fresh dill. Season with salt and pepper to taste.
Serve the soup warm with a dollop of sour cream and a sprinkle of fresh dill.
