Microwave cubed squash (about 4 1/2 cups worth) for 5-7 minutes until fork tender.
Dice onion and mince garlic.
Add oil to a stockpot and heat over medium.
Add squash, onion, and garlic to the hot oil; sauté for 7-8 minutes, stirring very little at first, then more frequently until squash starts to turn golden brown.
Reduce heat to low and add sugar; continue cooking about 10 minutes more until all vegetables are a rich, spotty, caramel color.
Add in cinnamon, nutmeg, ginger, cloves, and cayenne pepper; continue to sauté 30-60 seconds longer until fragrant.
Stir in broth, then bring to a low boil over medium-high heat.
Reduce heat to low and simmer, partially covered, for about 10 minutes until squash is tender.
Use an immersion blender and puree 30-60 seconds until very smooth. (If using a traditional blender, vent it either by removing the lid's pop-out center or by lifting one edge of the lid. Drape the blender canister with a kitchen towel. To 'clean' the canister, pour in a little half-and-half, blend briefly, then add to the soup.)
Return soup to the pot and stir in half-and-half. Season with salt and pepper to taste. (If soup seems too thick, add a little more broth until desired consistency is reached.)
Cook until heated through. Serve with apple slices on the side.