Flatten steaks to ⅓-inch thickness using a rolling pin.
Slice steaks into ⅕-inch strips, discarding excess fat.
Using salt and pepper, season the beef.
Over high heat, heat 1 tablespoon of oil.
Cook half of the beef until browned, then set aside.
Add another tablespoon of oil to the remaining beef.
Reduce heat to medium-high. Add butter, onions, and cook for 1 minute.
Add mushrooms, cook until golden, and scrape the pan.
Add flour and cook for about 1 minute.
Stir in sour cream and mustard while gradually adding broth.
Reduce the heat to medium-low and simmer the mixture.
Once the stroganoff sauce thickens (about 3 to 5 minutes), adjust salt and pepper.
Add beef and its juices to the sauce; simmer for 1 minute.
Serve over pasta or egg noodles after removing them from the stove.
Garnish with chives if desired.