Preheat the oven to 350°F. Bring a large pot of salted water to a boil and cook the egg noodles until just al dente. Drain and set aside.
In a large skillet over medium heat, cook the ground beef, breaking it up as it browns. Add the diced onion, sliced mushrooms, and minced garlic. Season with salt and pepper and cook until the vegetables soften.
Push the beef mixture to one side of the skillet. Add the butter to the empty space. Once melted, sprinkle in the flour and stir for one minute to form a roux.
Slowly pour in the beef broth while stirring, then add the Worcestershire sauce. Bring the mixture to a simmer and cook until it thickens slightly.
Remove the skillet from the heat and stir in the sour cream until the sauce is smooth. Add the drained noodles and gently toss everything together.
Transfer the mixture to a greased 9x13 baking dish. Sprinkle the shredded cheddar cheese evenly over the top.
Bake for 20 minutes, or until the cheese is melted and the edges are bubbling. Let it rest for 5 minutes before serving.