In a large pot or Dutch oven on the stovetop over medium heat, melt the butter. Add chopped onions and sauté until they become translucent.
Stir in the sliced mushrooms and cook until they have softened and released their moisture, stirring occasionally.
Sprinkle flour over the mushrooms and onions, stirring well to coat everything evenly. Cook for an additional 1 minute. Pour in the broth and stir to combine. Reduce the stovetop heat to low and simmer for 10 minutes, stirring occasionally.
While soup simmers, in a separate bowl, whisk together the sour cream and milk until smooth.
After simmering for 10 minutes, stir in the paprika, dill, salt, thyme, and pepper. Gradually add the dairy mixture to the soup, stirring continuously to incorporate it well. Increase stovetop heat to medium-low and simmer the soup for an additional 5-7 minutes. Check the seasoning and add more, if desired.
Ladle the soup into bowls, garnish with chopped parsley and pepper, then serve.