Melt butter in a heavy skillet over medium heat; stir in onion and mushrooms. Reduce heat; cook and stir until onions release their liquid and mushrooms soften, about 5 minutes.

Season chicken strips with Greek seasoning and black pepper. Place chicken in the skillet; cook and stir over medium heat until lightly browned on all sides, about 5 minutes.

Meanwhile, bring a large pot of lightly salted water to a boil. Cook noodles at a boil until tender yet firm to the bite, about 10 minutes; drain.

Pour chicken broth, milk, and Worcestershire sauce into skillet; slowly stir in sour cream, a tablespoonful at a time. Spoon out a small amount of sauce and whisk with flour in a small bowl until smooth. Stir flour mixture back into sauce; simmer until sauce thickens, about 5 minutes. Season with salt and ground black pepper. Serve over egg noodles; garnish with parsley.