Prepare the cauliflower by cutting around the core with a paring knife to remove it; slice the core thinly and set aside. Measure out a heaping cup of small half-inch florets and reserve for the garnish. Slice the remaining head into half-inch thick pieces.

Melt a portion of the butter in a large saucepan over medium-low heat. Add the leek, onion, and salt, cooking until softened but not browned, about 7 minutes.

Stir in the water, the sliced core, and half of the sliced cauliflower head. Increase the heat to medium-high and bring to a simmer, then reduce to medium-low and simmer gently for 15 minutes. Add the remaining sliced cauliflower and continue to simmer until the cauliflower is completely tender and crumbles easily, 15 to 20 minutes.

While the soup simmers, melt the remaining butter in a small skillet over medium heat. Add the reserved small florets and cook, stirring frequently, until they are golden brown and the butter has a nutty aroma and turns a deep amber, 6 to 8 minutes. Remove from the heat, use a slotted spoon to transfer the florets to a bowl, and toss them with vinegar; pour the browned butter into a separate bowl for drizzling.

Puree the soup in a blender until completely smooth, about 45 seconds. Return it to a clean pot and bring to a brief simmer over medium heat, adjusting the texture with a splash of extra water if needed—it should be thick and velvety enough to settle into a flat surface after stirring. Season with salt to taste and serve hot, garnishing each bowl with the browned florets, a drizzle of the reserved butter, chives, and pepper.