Marinate chicken in Italian dressing and Greek seasoning in a bowl, at least 30 minutes.
Place chicken in a 9x13-inch baking dish. Sprinkle top with additional Greek seasoning.
Bake in preheated oven until no longer pink in center and juices run clear, about 25 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Melt butter in a skillet over medium-high heat until bubbling. Pour in wine, Worcestershire sauce, and garlic salt; bring to a boil. Stir in mushrooms; reduce to a simmer, cover, and cook until mushrooms are tender, about 3 to 5 minutes. Seasoning mixture with salt.
Arrange green chili peppers lengthwise and Havarti cheese on top of chicken. Return to the oven and bake until cheese has melted, about 5 minutes. Top chicken with mushrooms and pan juices.
