Cook pasta according to package instructions.
While pasta is cooking add cooking oil to large skillet or saute pan and set over medium heat. Add onions, peppers, and mushrooms and saute until mushrooms are cooked through. Add garlic and cook an additional 2 minutes. Onions and peppers should be slightly tender but still retain some firmness.
Remove vegetables from the skillet, add additional cooking oil as needed, then add the sliced beef, salt, and black pepper. Sear over medium high heat until the meat is no longer pink.
Remove beef from the skillet, and shut off heat to let pan cool slightly. Once the pan is cool enough not to scorch butter, prepare the sauce by melting butter in the skillet over low heat and sprinkle in flour to form a roux. Stir continually to prevent burning and cook the roux for 3-5 minutes, or until beginning to brown. (you want to cook it long enough to cook out the raw flour taste)
Slowly incorporate milk, beef broth, and Worcestershire a little bit at a time, stirring the sauce to fully incorporate before more liquid is added to prevent lumps.
Add shredded cheese in batches, stirring to melt into the sauce.
Place noodles in a large bowl with beef and vegetables and pour sauce over. Mix to combine everything and serve.