Heat oil over medium-high heat in large, deep sauté pan until simmering. Add beets, then onion; saute, stirring very little at first, then more frequently, until onion starts 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 caraway seeds 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 beets are tender, about 10 minutes.
Using an immersion blender or traditional blender, puree (adding fresh dill) until very smooth, 30 seconds to 1 minute.
Return to pan; 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.
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