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, 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 more. Pour in the broth and 1 cup water and bring to a simmer.
Add the corn and fennel and simmer until the vegetables 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. Remove and discard the thyme and bay leaf 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).
Stir in the heavy cream and reheat the soup if necessary. Adjust the consistency and seasoning with water, salt and pepper.
Serve hot sprinkled with corn kernels and fennel fronds.
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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