Pull off the outer leaves of the cauliflower and trim the stem. Using a paring knife, cut around the core to remove it; thinly slice the core and reserve. Cut a heaping cup of small half-inch florets from the head and set aside for the garnish. Cut the remaining cauliflower crosswise into half-inch-thick slices.

Melt butter in a large saucepan over medium-low heat. Add the leek, onion, and salt. Cook, stirring often, until the vegetables are softened but not browned, about 7 minutes.

Add the water, the sliced cauliflower core, and half of the sliced cauliflower. Increase the heat to medium-high and bring to a simmer, then reduce the heat 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 an 8-inch skillet over medium heat. Add the reserved small florets and cook, stirring often, until the florets are golden brown and the butter is browned and smells nutty, 6 to 8 minutes. Use a slotted spoon to transfer the florets to a small bowl, toss them with the vinegar and salt to taste, and reserve the browned butter in a separate bowl.

Process the soup in a blender until completely smooth, about 45 seconds. Return the pureed soup to a clean pot and bring to a brief simmer over medium heat. Adjust the consistency with the remaining water as needed until the soup is thick and velvety but thin enough to settle with a flat surface after being stirred. Season with salt to taste and serve hot, garnishing each bowl with the browned florets, a drizzle of the reserved brown butter, chives, and pepper.