Heat the oil in a medium saucepan over medium heat. Add the onion and cook, stirring, until tender, 3 to 5 minutes. Add the potato, thyme sprigs, 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 simmer.
Add the asparagus, peas and zucchini and simmer until the potatoes and vegetables are tender, about 20 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 in the basil 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 per batch).
Reheat the soup if necessary and adjust the consistency and seasoning with additional water, salt and pepper.
Serve hot sprinkled with asparagus tips, peas and thyme leaves.
When blending hot liquid, first let it cool for 5 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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