Melt butter in a large skillet over medium heat. Whisk in cream cheese, Gouda cheese, mayonnaise, chicken broth, and ranch dressing mix. Reduce heat to low and cook, stirring constantly, until completely incorporated, 4 to 5 minutes. Stir in broccoli, chicken, salt, and pepper until combined. Transfer to a casserole dish and top with cheddar cheese.
Bake in the preheated oven until cheese is melted and the broccoli-chicken mixture is heated through about 25 minutes. Switch the oven to broil, and cook until cheese starts to brown, 2 to 4 minutes.