Pound the steak pieces with salt and black pepper. Dust the steak with the flour, coating both sides evenly. Make all excess flour fall away.
Heat 1 tablespoon of oil in a large skillet over medium-high heat. Working in batches, brown the beef on both sides until browned, about 3 to 5 minutes. Season the mushrooms with salt and black pepper. Transfer the beef to a bowl.
Add the wine to the skillet and bring to a boil, scraping up the browned bits from the bottom of the pan. Simmer until reduced by half.
Melt the butter in the skillet and add the mushrooms and onion. Cook for 3 to 5 minutes until the mushrooms begin to release their liquid and the onion becomes translucent.
Return the beef and any accumulated juices to the pan. Add the beef broth and Worcestershire sauce and bring to a simmer. Cook until the sauce thickens and reduces slightly.
Remove from heat and stir in the sour cream. Season to taste with salt and black pepper.
To serve, spoon the beef stroganoff over the cooked noodles and rice with sauce.