Heat oil over medium-high heat in a large, deep sauté pan until shimmering.
Add sweet potatoes, then onion; sauté, stirring very little at first, then more frequently, until vegetables start to turn golden brown, 7 to 8 minutes.
Reduce heat to low and add butter, sugar and garlic; continue cooking until all vegetables are a rich spotty caramel color, about 10 minutes longer.
Add ginger, nutmeg and cayenne pepper; continue to sauté 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 sweet potatoes are tender, about 10 minutes.
Using an immersion blender or traditional blender, puree until very smooth, 30 seconds to 1 minute.
Return to pan (or a soup pot); add enough half-and-half so the mixture is souplike, yet thick enough to float garnish. Taste, and add salt and pepper if needed. Heat through, ladle into bowls, garnish and serve.