Trim the outer leaves from the head of cauliflower. Trim around the core and remove to a cutting board. Remove the tough outer layer of the stem then slice thin. Set aside.
Cut enough florets (½-inch pieces) to generously fill a 1 cup measure. Set aside. Cut the remaining cauliflower into ½ inch thick slices.
In a large saucepan or Dutch oven, add 3 tablespoons butter. Heat over medium-low heat and add the leek, onion and 1 ½teaspoons salt. Cook, stirring frequently under the leek and onion are tender but not brown.
Turn the heat up to medium-high and add 5 cups of water, the sliced core and half the sliced cauliflower florets. Bring the soup to a boil, then reduce the heat to a simmer. Cook about 20 minutes or until the vegetables are tender. Add the remaining sliced cauliflower, return to simmer and continue to cook until the cauliflower is tender and crumbles easily, 15-20 minutes longer.
While the soup is cooking, melt 5 tablespoons butter in a small skillet over medium heat. Add the reserved 1 cup of florets and cook until they are golden brown and the butter solids are starting to brown. Turn the florets frequently until done, then remove to a bowl using a slotted spoon. Drizzle the browned florets with the balsamic vinegar and generously season with salt and pepper. Do not discard the browned butter as you will use it to garnish individual bowls of soup.
One the soup vegetables are tender, carefully transfer the hot mixture to a blender. Puree the soup until completely creamy - about 1 minute, depending on the power of your blender. Pour the soup back into the pot and heat just until simmering. Add a little water if needed to make a smooth creamy soup. You don't want it to look like runny mashed potatoes, or be too thick.
Serve in individual bowls garnished with the browned florets, a drizzle of browned butter, salt and pepper to taste and fresh chopped chives.