Melt butter in large pan over medium het. Add onions and saute til tender and aromatic, 5 to 7 minutes.
Sprinkle flour and dry mustard over onions and stir to evenly coat.
Continue cooking, stirring often for 1 to 2 minutes. Add broth, stirring til well blended.
Peel, quarter and core 1 of the apples and finely chop it. Add apple to the soup, cover, reduce the heat to low and simmer til soup is thickened and the apple is tender, about 10 minutes.
Stir in cider and grated cheese and cook a few minutes longer, stirring constantly, til cheese is fully melted. Puree soup in a blender. Stir in a few drops of hot pepper sauce and salt and pepper to taste.
Quarter and core the remaining apple and cut in 1/2 inch dice. Ladle soup into warmed bowl, sprinkle with diced apple and serve at once. Serves 4 to 6