3 tablespoons unsalted butter, divided
8 ounces mushrooms, stems trimmed and sliced 1/4 inch thick
1 medium yellow onion, small dice
2 medium garlic cloves, finely chopped
1 pound lean ground beef
3/4 teaspoon salt, plus more for seasoning
1/4 teaspoon freshly ground black pepper, plus more for seasoning
3 tablespoons all-purpose flour
1 1/4 teaspoons paprika
4 cups (1 quart) beef broth
1 tablespoon Dijon mustard
8 ounces dried egg noodles
3/4 cup whole milk Greek yogurt or sour cream
1 tablespoon coarsely chopped fresh Italian parsley leaves (optional)
Melt 1 tablespoon of the butter in a large, heavy-bottomed pot or Dutch oven over medium-high heat. Add the mushrooms, season with salt and pepper, and cook, stirring rarely, until browned, about 5 minutes. Transfer to a medium bowl and set aside.
Reduce the heat to medium and add the remaining 2 tablespoons butter. When the butter has melted, add the onion and garlic, season with salt and pepper, and cook, stirring occasionally, until softened, 4 to 5 minutes. Add the beef, 3/4 teaspoon salt, and 1/4 teaspoon pepper. Cook, breaking the meat into smaller pieces with a wooden spoon and stirring occasionally, until just cooked through and no longer pink, 6 to 8 minutes.
Sprinkle in the flour and paprika. Cook, stirring frequently, until the raw flavor has cooked off the flour, about 1 to 2 minutes. Add the broth and mustard and scrape up any browned bits from the bottom of the pot. Increase the heat to medium-high and bring to a simmer.
Add the noodles and reserved mushrooms along with their accumulated juices and stir to combine. Reduce the heat to medium-low and simmer, stirring occasionally and making sure the noodles are always mostly submerged in liquid, until the noodles are just cooked through and tender, about 6 to 8 minutes.
Remove from the heat and stir in the yogurt or sour cream until completely incorporated. Taste and season with salt and pepper as needed. Sprinkle with the parsley if desired and serve immediately.