Heat 2 tablespoons of the oil in a medium saucepan over medium heat. Add the onion and cook, stirring, until soft, 3 to 5 minutes. Add the potato, garlic, cayenne, nutmeg, 1/2 teaspoon salt and a few grinds of pepper and cook, stirring, until the potatoes are well coated, 2 to 3 minutes. Pour in 3 1/2 cups of water and bring to a simmer.
Add the sweet potato, bring back to a simmer and cook until both potatoes are tender, about 20 minutes, adjusting the heat as needed to maintain a simmer. Remove from the heat and let cool at least 5 minutes. Carefully puree until smooth with a handheld immersion blender or in 2 to 3 batches in a blender (do not fill the blender more than halfway full).
Meanwhile, combine the remaining 2 tablespoons of oil and sliced garlic clove in a small skillet and heat over medium. Swirl the pan as the garlic becomes fragrant and turns golden brown, 4 to 5 minutes. Remove the toasted garlic from the oil to a paper towel-lined plate. Reserve with garlic oil.
Reheat the soup if necessary, and adjust the consistency and seasoning with additional water, salt and pepper. Stir in the butter off the heat right before serving.
Serve hot and garnish with the toasted garlic, a drizzle of garlic oil and a sprinkle of parsley.
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When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.