Melt butter in a large saucepan over medium heat; add onions and red pepper. Sauté until onions are soft and golden. Stir in flour and mustard; continue cooking for an additional minute or two. Stir in chicken broth, applesauce, sugar and thyme. Bring to a boil. Lower the heat and simmer covered for 10 minutes.
Puree mixture in a blender or food processor and return to saucepan (or omit the puree step for a slightly less smooth soup.) Stir cheese into soup base and heat until cheese is fully melted. Add salt to taste.
Core (do not peel) the apple and cut into ½-inch pieces. Ladle soup into bowls and top with apple pieces then garnish with a sprig of thyme. Yields 8 cups.