Remove the outer leaves from the cauliflower. Trim the core, peel away the tough outer layer, and slice it thinly.
Cut about 1 cup of small cauliflower florets and set them aside. Slice the remaining cauliflower into 1/2-inch pieces.
In a large pot or Dutch oven, melt 3 tablespoons of butter over medium-low heat.
Add the sliced leek, onion, and about 1 1/2 teaspoons of salt. Cook gently, stirring often, until softened but not browned.
Increase the heat to medium-high and add 5 cups of water, the sliced core, and half of the cauliflower pieces. Bring to a boil.
Reduce the heat to a simmer and cook for about 20 minutes, until the vegetables begin to soften.
Add the remaining cauliflower, return to a simmer, and cook another 15–20 minutes until everything is very tender.
While the soup cooks, melt the remaining 5 tablespoons of butter in a skillet over medium heat.
Add the reserved cauliflower florets and cook, stirring often, until golden brown and slightly crisp, with the butter turning nutty and browned.
Remove the florets with a slotted spoon, drizzle with white balsamic vinegar, and season with salt and pepper. Reserve the browned butter.
Carefully blend the soup until completely smooth using a blender or immersion blender.
Return the soup to the pot and heat gently. Add a little extra water if needed to reach a smooth, pourable consistency.
Taste and adjust seasoning with salt and pepper.
Serve warm, topped with the browned cauliflower florets, a drizzle of browned butter, and fresh chives.