Real Cream of Mushroom Soup

6 T. salted butter, divided
1 pound sliced fresh button mushrooms
1 small onion, chopped
1 tsp. kosher or sea salt, plus more to taste
1/4 tsp. freshly ground black pepper
3 T. all-purpose flour
1 cup chicken broth
2 cups whole milk or half-and-half
1-1/2 cups sour cream, at room temperature
Garnish: chopped fresh parsley

Melt 3 T. butter in a 4- to 5-quart heavy-bottomed pot over medium-high heat. Add mushrooms, onion, salt, and pepper, and cook, stirring often, until golden brown, about 15 minutes. Transfer to a bowl and set aside. Reduce heat to medium and melt remaining 3 T. butter. Add flour and cook, stirring, until glossy, about 3 minutes. Slowly drizzle in chicken broth and milk (or half-and-half), whisking as you go. Increase heat to a simmer and cook until mixture thickens, whisking often, about 5 minutes. Reduce heat to medium-low, stir in the reserved mushroom mixture, and cook 10 minutes, stirring occasionally. Remove from heat and add the sour cream. Serve garnished with chopped fresh parsley.