Trim and quarter 12 ounces baby bella, cremini, or button mushrooms (cut into eighths if large). Halve and thinly slice 1 large yellow onion (about 1 1/2 cups). Finely chop 2 garlic cloves.
Cut 1 1/2 pounds top sirloin, boneless ribeye, beef tenderloin, or flank steaks across the grain into 1/2-inch-thick slices. If the slices are long, cut crosswise into 1-inch-long pieces. Place in a large plastic bag. Using the flat side of a meat pounder or rolling pin, lightly pound the beef to flatten slightly and tenderize. Add 2 tablespoons of the all-purpose flour, 3/4 teaspoon hot paprika, and 1 teaspoon kosher salt. Seal the bag and shake until the beef is evenly coated.
Heat 1 tablespoon neutral oil in a large skillet over high heat until shimmering. Working in batches, add the beef slices in a single layer and sear until browned on the outside but still rare inside, 30 to 60 seconds per side, reducing heat as needed to prevent scorching. Transfer to a plate. Sprinkle the seared beef with any remaining flour mixture from the bag.
Reduce the heat to medium-high and add 1 tablespoon of the unsalted butter to the pan. Once melted, add the onions and cook, stirring occasionally, until starting to soften and turn translucent, about 4 minutes. Add the mushrooms and a pinch of kosher salt. Cook, stirring frequently, until the mushrooms and onions are deeply browned, 10 to 15 minutes.
Add the garlic, remaining 1 tablespoon all-purpose flour, and remaining 2 tablespoons unsalted butter. Stir until the butter is melted and the mixture is fragrant, about 30 seconds. Add 1 (14.5-ounce) can low-sodium beef or chicken broth and bring to a simmer. Cook until reduced by half, about 5 minutes.
Reduce the heat to medium-low. Add 2/3 cup sour cream, 1 tablespoon Dijon mustard, and 1 1/2 teaspoons Worcestershire sauce. Stir until the sour cream is fully incorporated (don’t worry if it looks broken or separated at first). Return the beef and any accumulated juices on the plate to the pan. Cook, stirring occasionally, until the beef is medium-rare, about 3 minutes.
Remove the pan from the heat. Taste and season with kosher salt and freshly ground black pepper as needed. Garnish with 2 tablespoons finely chopped parsley or chives if desired. Serve over cooked egg noodles, polenta, or mashed potatoes.