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1
Heat oil over medium-high heat in a large, deep saute pan until shimmering.
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2
Add potatoes, then onion; saute, stirring very little at first, then more frequently, until squash start to turn golden brown, 7 to 8 minutes.
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3
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.
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4
Add cayenne pepper; continue to saute until fragrant, 30 seconds to 1 minute longer.
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5
Add broth; bring to a simmer over medium-high heat. Reduce heat to low and simmer, partially covered, until potatoes are tender, about 10 minutes.
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6
Using an immersion blender or traditional blender, puree (adding fresh rosemary)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.)
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7
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.
