Microwave squash until fork tender; about 5-7 minutes.
Add oil to a stockpot, heat on medium.
Add butternut squash, then onion and garlic; saute, stirring very little at first, then more frequently, until squash start to turn golden brown, 7 to 8 minutes.
Reduce heat to low and add sugar; continue cooking until all vegetables are a rich spotty caramel color, about 10 minutes longer.
Add cinnamon, ginger, cloves, and cayenne pepper; continue to saute until fragrant, 30 seconds to 1 minute longer.
Add broth; bring to a simmer over medium-high heat. Reduce heat to low and simmer, partially covered, until squash is tender, about 10 minutes.
Using an immersion blender or traditional blender, puree until very smooth, 30 seconds to 1 minute. (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.)
If soup seems too thick, add a little more broth.
Return to pan (or a soup pot); add enough half-and-half so the mixture is soup-like, yet thick enough to float garnish. Taste, and add salt and pepper if needed. Heat through, ladle into bowls, garnish and serve.