Soup (without sour cream and toppings) can be made 3 days ahead. Let cool; cover and chill. Reheat before adding sour cream as directed.
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1. Prepare your ingredients. Chop **4 medium yellow or white onions (about 1½ lb.)** into ¼" pieces. (You should have about 4 cups.) Gently clean **2 lb. mushrooms (crimini and/or shiitake)** with a kitchen towel or paper towels. Remove and discard stems; slice caps ¼" thick. Peel **2 medium carrots (about 12 oz.)**. Cut in half lengthwise, then cut those halves in half lengthwise (you’ll have 4 long spears per carrot). Cut crosswise into ¼"-thick pieces.
 
2. Smash, peel, and finely chop **6 garlic cloves**. Finely chop **2 handfuls of parsley leaves with tender stems**until you have ¼ cup; set parsley aside.
 
3. Melt **½ cup (1 stick) unsalted butter** in a medium Dutch oven or other heavy pot over medium heat. Add onion and a pinch of kosher salt (this will help the onion release water and cook more quickly) and cook, stirring occasionally with a wooden spoon, until onion is softened and browned, about 10 minutes. Add mushrooms and carrots to pot and cook, stirring often, until mushrooms start to brown, 12–16 minutes. Add garlic, **4 tsp. Hungarian sweet paprika**, **4 tsp. dried dill or ¼ cup fresh dill**, and **2 tsp. dried thyme or 2 Tbsp. fresh thyme** and cook, stirring often, until fragrant, about 2 minutes.
 
4. Add **6 cups low-sodium vegetable broth** and bring to boil over high heat. Reduce heat to medium-low and simmer, uncovered, 10 minutes.
 
5. Whisk **2 cups whole milk** and **¼ cup all-purpose flour** in a small bowl or measuring glass to combine. Pour into soup and stir well. Bring to a boil over high heat. Reduce heat to medium-low and simmer until soup is thick enough to thinly coat spoon, about 8 minutes. Remove from heat.
 
6. Mix **¾ cup sour cream** and 2–3 Tbsp. broth from soup until loose, then add to soup and stir well to incorporate. Season with salt and **freshly ground black pepper**.
 
7. Ladle soup into bowls and top each with a dollop of sour cream, some reserved parsley, and a pinch of paprika.