Grilled cheese day was always my favorite in elementary school. They'd serve up a big, hearty grilled cheese sandwich with a cup of vegetable soup. Both were amazing, which is not something you usually say about school cafeteria food. Then you realize that grilled cheese and soup are the two easiest things to make. No wonder they were great. You only have to worry when the school cafeteria starts to serve up crab rangoon.
Give me anything hot or cold to dip my grilled cheese in. Applesauce sounds just as good as anything else. People eat cheese and apples all the time together. Sounds even better when one is fried and the other is mashed. Just don't mash the grilled cheese and fry the applesauce. It will taste horrible and I doubt it'll be good.