Add the dressing after your mayonnaise or butter so the bread doesn't get soggy. Using a very thin layer is important because you don't want to soak the bread.
Once you have both pieces of bread all dressed and ready to go, cook your sandwich the way you normally would in a saute pan or using a sandwich press.
Vinaigrette dressings like Italian and balsamic vinaigrette are usually a safe bet on your grilled cheese. Creamy dressings also work well because most are made with mayonnaise.
For example, pair an apple cider vinaigrette on a grilled apple and brie sandwich or spread green goddess dressing onto a sandwich filled with cheddar cheese and spinach.