Bring the stock and water to a simmer in a medium saucepan.
Heat 2 tablespoons of the oil in a large enameled cast-iron Dutch oven over medium heat until shimmering. Add the fennel, celery, shallots, garlic and a generous pinch each of salt and pepper. Cook, stirring occasionally, until the vegetables just start to soften but not brown, 5 to 7 minutes.
Meanwhile, peel, quarter and thinly sliced the potatoes. Add them to the Dutch oven along with the hot stock. Cover and simmer over medium heat until the vegetables are tender, about 10 minutes.
Cook the pancetta in a large skillet over medium heat until crisp, about 4 minutes. Transfer with a slotted spoon to a paper towel-lined plate to drain. Add the remaining 2 tablespoons oil and the crushed fennel seeds to the skillet. Cook over medium heat, stirring, until fragrant, about 1 minute. Remove from the heat.
Working in batches, puree the soup in a blender or food processor until smooth. (See Cook's Note) Alternatively, use an immersion blender to puree the soup. Return the soup to the Dutch oven and season with salt and pepper.
Ladle the soup into bowls and top with the crispy pancetta, Cheddar, snipped chives and sour cream, if using, and drizzle with the fennel oil. Serve.
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.