Melt Butter and Sauté Onions: In a large pot or Dutch oven on the stovetop over medium heat, melt butter. Add chopped onions and sauté until they become translucent, about 3 to 4 minutes.
Cook Mushrooms: Stir in sliced mushrooms and cook until they have softened and released their moisture, about 5 to 6 minutes, stirring occasionally.
Add Flour: Sprinkle flour over mushrooms and onions, stirring well to coat everything evenly. Cook for an additional 1 minute.
Pour and Simmer Broth: Pour in broth, stir to combine, and bring to a gentle simmer. Reduce stovetop heat to low and simmer for 10 minutes, stirring occasionally.
Prepare Dairy Mixture: While soup simmers, in a separate bowl, whisk together sour cream and milk until smooth.
Add Spices and Dairy: After simmering for 10 minutes, stir in paprika, dill, salt, thyme, and pepper. Gradually add dairy mixture to the soup, stirring continuously to incorporate it well. Increase stovetop heat to medium-low and simmer soup for an additional 5 to 7 minutes, until hot and slightly thickened (avoid boiling to prevent curdling).
Check Seasoning: Taste and add more salt and pepper, if desired.
Garnish and Serve: Ladle soup into bowls, garnish with chopped parsley and pepper, then serve.