Heat the oil in a medium saucepan over medium heat. Add the leeks and cook, stirring, until soft, 3 to 5 minutes. Add the potato, thyme, bay leaf, garlic, 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 the broth and bring to a simmer.
Add all but 2 to 3 of the broccoli florets, bring back to a simmer and cook until the broccoli is tender, about 20 minutes, adjusting the heat as needed to maintain a simmer. Remove from the heat, and let cool at least 5 minutes. Remove and discard the thyme and carefully puree until smooth with a handheld immersion blender or in 2 to 3 batches in a blender (do not fill blender more than halfway full).
Meanwhile, cut the reserved broccoli florets in half, lengthwise. Put in a small microwave-safe bowl, toss with a splash of water and a pinch of salt, cover and cook until bright green and crisp-tender, 2 to 3 minutes. Set aside.
Add the cream and reheat the soup if necessary. Adjust the consistency and seasoning with water, salt and pepper.
Serve hot and garnish each soup with a broccoli half.
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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