Pull off outer leaves of cauliflower and trim stem. Using a paring knife, cut around core to remove; thinly slice core and reserve. Cut heaping 1 cup of 1/2-inch florets from the head of cauliflower; set aside. Cut remaining cauliflower crosswise into 1/2-inch thick slices.
Melt 3 tablespoons butter in a large saucepan over medium-low heat. Add leek, onion, and 1 1/2 teaspoons salt; cook, stirring frequently until leek and onion are softened but not browned, about 7 minutes.
Increase heat to medium-high; add 4 1/2 cups water, sliced core, and half of the sliced cauliflower; and bring to simmer. Reduce heat to medium-low and simmer gently for 15 minutes. Add remaining sliced cauliflower, return to simmer, and continue to cook until cauliflower is tender and crumbles easily, 15 to 20 minutes longer.
While soup simmers, melt the remaining 5 tablespoons butter in 8-inch skillet over medium heat. Add reserved florets and cook, stirring frequently, until florets are golden brown and butter is browned and imparts nutty aroma, 6 to 8 minutes. Remove skillet from heat and use a slotted spoon to transfer florets to a small bowl. Toss florets with vinegar and season with salt to taste. Pour browned butter in skillet into small bowl and reserve for garnishing.
Process soup in a blender until smooth, about 45 seconds. Rinse out the pan. Return pureed soup to pan and return to simmer over medium heat, adjusting consistency with remaining water as needed (soup should have thick, velvety texture but should be thin enough to settle with a flat surface after being stirred) and seasoning with salt to taste. Serve, garnishing individual bowls with browned florets, a drizzle of browned butter, and chives and seasoning with pepper to taste.