Sauté the Vegetables: Melt the butter in a heavy skillet over medium heat. Add the diced onion and sliced mushrooms, then reduce the heat. Cook and stir for about 5 minutes, until the onions release their liquid and the mushrooms have softened. Season the Chicken: Sprinkle the thin chicken strips evenly with the Greek seasoning and freshly ground black pepper.
Brown the Chicken: Add the seasoned chicken to the skillet with the vegetables. Cook and stir over medium heat for about 5 minutes, until the chicken is lightly browned on all sides.
Cook the Egg Noodles: While the chicken cooks, bring a large pot of lightly salted water to a boil. Add the egg noodles and cook at a boil for about 10 minutes, or until tender but still firm to the bite. Drain well.
Build the Sauce: Pour the chicken broth, milk, and Worcestershire sauce into the skillet with the chicken and vegetables. Stir gently to combine.
Add the Sour Cream: Slowly incorporate the sour cream into the skillet, adding it about a tablespoonful at a time and stirring as you go to blend it smoothly into the sauce.
Prepare the Thickener: Spoon a small amount of the sauce into a small bowl. Add the flour and whisk until the mixture is smooth and free of lumps.
Thicken the Stroganoff: Stir the flour mixture back into the skillet sauce. Allow the sauce to simmer for about 5 minutes, stirring as needed, until it thickens.
Adjust the Seasoning: Taste the finished sauce and season with salt and additional ground black pepper as needed.
Serve: Spoon the creamy chicken and mushroom stroganoff over the drained egg noodles and garnish with chopped fresh parsley before serving.