Cut the Delicata squash in half lengthwise. Salt, pepper, oil, and squeeze a little lemon juice on the cut sides. Place on a small oiled roasting pan. Roast in a 375-degree oven until soft when pricked with a knife, and the cuts sides are nicely caramelized. This takes about 30 minutes. When the squash is roasted to your liking, scoop out the seeds. Then scoop out the squash flesh and place in a 2-quart saucepan. Mash the squash.
Add the stock to the saucepan, and blend thoroughly. Bring to a simmer, and then add the grated apple. After a few minutes, add the vermouth. After a few minutes more, season with salt and pepper. Finally add the cheese. Do not allow soup to violently boil.
After about 20 minutes, take the soup off the stove. Either allow the soup to cool thoroughly before you put it through a blender, or divide the warm soup into 2 portions and run them individually. The last thing you want is a full blender of hot soup exploding all over your kitchen!
Serve immediately, or wait a day to let the flavors combine well. Top the soup off with your choice of possible additions - more cheese, seeds, nuts, Dukkah, etc. Serve with a nice bread and a salad for a great lunch.
NB: you can make vegan by using vegetable stock and omitting cheese. Add seasoned nuts or seeds to add flavor.