Lower the heat to medium and add 1 tablespoons unsalted butter, divided and lay 8 oz cremini mushrooms, sliced. Cook undisturbed for 3 to 4 minutes until deep golden brown. Flip the mushrooms and brown the other side for another 3 to 4 minutes. Push the mushrooms to one side of the pan, add ½ medium onion, diced to the cleared space, and cook for 3 to 4 minutes until softened. Stir in 1 garlic cloves, minced and cook with the onions for 1 to 2 minutes until fragrant.
Spoon 2 tablespoons tomato paste directly over the onions. Cook, stirring the paste and onions continuously, for 1 to 2 minutes until the paste darkens to a deep red and smells sweet and roasted. Fold the mushrooms back into the onion mix,using your spatula to scrape up the browned fond stuck to the bottom of the pan.
Sprinkle 1 tablespoon all-purpose flour evenly over the caramelized tomato paste and vegetable mixture. Stir constantly for 1 minute to toast the starch and cook out any raw flour taste.
Gradually pour in t1½ cups low-sodium chicken broth while whisking continuously to dissolve the seared paste and toasted flour into a smooth liquid. Bring to a gentle simmer, scraping up any remaining caramelized bits from the skillet bottom. Reduce the heat to medium-low. Let the sauce simmer gently for about 3 minutes, stirring occasionally, until it thickens and coats the back of a spoon.
Turn the burner to its lowest setting. In a small bowl, whisk a splash of the warm pan broth directly into ½ cup full-fat sour cream to temper it. Fold the tempered sour cream and 2 teaspoons Dijon mustard into the skillet, stirring gently until the gravy becomes completely smooth. Return the cooked chicken into the pan, stirring to coat the meat and let it warm back up. Remove the skillet from the heat, drop in 1 tablespoons unsalted butter, divided, and swirl or whisk gently until fully melted and glossy. Garnish with finely chopped fresh parsley, and serve immediately over hot buttered mashed potatoes or broad egg noodles.