Heat 2 tablespoons of the oil in a medium saucepan over medium heat. Add the onion and cook, stirring, until tender, 3 to 5 minutes. Add the thyme sprigs, bay leaf, cayenne, 1 cup of the potato, 1/2 teaspoon salt and a few grinds of black pepper and cook, stirring, until the potatoes are well coated, 2 to 3 minutes more. Pour in the broth and 1 1/2 cups water and bring to a simmer.
Reserve two florets (about 2 inches in diameter) of the cauliflower. Add the fennel and the remaining cauliflower 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 stir in the parsley. 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).
Meanwhile, place the remaining 1/4 cup potatoes in a microwave-safe bowl. Cover with plastic wrap and heat in the microwave for 4 minutes. Carefully unwrap and set aside to cool.
Slice the reserved cauliflower florets thinly, about 1/8 inch thick. (You need about 12 pieces.) Heat 1 tablespoon of the remaining oil in a small skillet over medium-high heat. Add the cauliflower pieces in a single layer and cook until golden brown on the bottom, about 2 minutes. Flip the pieces over and cook until tender and golden brown on the second side, 1 to 2 minutes more. Transfer to a paper-towel-lined plate to drain.
Add the remaining 1 tablespoon oil to the same skillet. Add the microwaved potato and season with salt. Cook, stirring, until golden brown all over, 8 to 10 minutes.
Reheat the soup if necessary. Adjust the consistency and seasoning with water, salt and pepper.
Serve hot sprinkled with some pan-fried cauliflower and potato and fresh parsley leaves.
Note: 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.
Copyright 2016 Television Food Network, G.P. All rights reserved.