Heat the oil in a medium saucepan over medium heat. Add the onion and cook, stirring, until soft, 3 to 5 minutes. Add the potatoes, thyme, garlic, 1/2 teaspoon salt and a few grinds of pepper and cook, stirring, until the potatoes are well coated, 2 to 3 minutes more. Pour in the broth and bring to a boil, then reduce the heat and simmer for 10 minutes.
Add the zucchini, bring back to a simmer and cook until tender, about 10 minutes, adjusting the heat as needed to maintain a simmer. Remove from the heat, discard the thyme and let cool at least 5 minutes. Stir the basil into the soup and 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).
Reheat the soup if necessary, adjusting the consistency and seasoning with additional water, salt and pepper.
Serve hot and garnish with a sprinkle of almonds, basil leaves and a drizzle of oil.
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.
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