Remove fat and gristle from roast; cut into strips 1/2-inch thick by 2-inches long. Season with 1/2 teaspoon salt and 1/2 teaspoon black pepper; set aside.

Melt 1 tablespoon butter in a large skillet over medium heat. Add mushrooms and green onions; cook, stirring occasionally, until mushrooms are browned, about 6 minutes. Transfer to a bowl.

Add 1 tablespoon butter to the same skillet. Add 1/2 beef strips; cook and stir until browned, about 5 minutes. Transfer to a separate bowl. Repeat with remaining 1 tablespoon butter and remaining 1/2 beef strips.

Pour wine into the hot skillet; scrape the browned bits of food off the bottom of the skillet with a wooden spoon.

Combine 1/4 cup beef broth and flour in a jar with a tight fitting lid; shake until combined. Stir into the skillet, whisking until smooth; stir in remaining broth and mustard, then return beef strips to the skillet. Bring to a simmer; cover and simmer until beef strips are tender, about 1 hour.