Flatten the steaks to about 3/4 cm thick using your fist, rolling pin, or mallet. Slice into thin strips and discard excess fat.
Season the beef lightly with salt and pepper.
Heat 1 tablespoon oil in a large skillet over high heat. Add half the beef in a single layer and sear quickly for about 30 seconds per side until browned. Remove to a plate.
Repeat with the remaining oil and beef.
Reduce heat to medium-high. Add butter, then sauté onions for 1 minute.
Add mushrooms and cook until golden brown, scraping up any browned bits from the pan.
Stir in flour and cook for 1 minute.
Gradually add half the beef broth while stirring, then add the remaining broth.
Add sour cream and Dijon mustard, stirring until combined.
Bring the Beef Stroganoff sauce to a gentle simmer and cook for 3–5 minutes until slightly thickened.
Return the beef and any juices to the skillet. Simmer for 1 minute only, then remove from heat.
Serve hot over pasta or egg noodles and garnish with chopped chives.