Flatten steak to about 0.75 cm thickness using a mallet or rolling pin. Slice into thin strips about 0.5 cm wide, trimming off excess fat. Season generously with salt and pepper.
Heat 1 tablespoon of vegetable oil in a large skillet over high heat. Add half the beef in a single layer and sear for 30 seconds per side, just until browned but still pink inside. Remove from the pan and repeat with the second half using the remaining oil.
Reduce heat to medium-high. Add butter to the same skillet, followed by the sliced onions. Cook for 1 minute, then add mushrooms. Sauté until mushrooms are golden and browned, scraping up any browned bits from the pan.
Sprinkle flour evenly over the mushrooms and stir for about 1 minute to cook off the raw taste. Gradually pour in half of the beef broth, stirring constantly until smooth. Then add the remaining broth, Dijon mustard, and sour cream. Stir and simmer on medium-low for 3 to 5 minutes until the sauce thickens to a silky, pourable consistency. Season with salt and pepper to taste.
Return the seared beef and its juices to the skillet. Simmer for 1 more minute until the beef is heated through.
Serve hot over pasta or egg noodles and garnish with chopped chives if desired.