Prepare the green garlic and leek. Slice off the root base of the green garlic and peel away any tough outer parts of the stalks. Rinse carefully under cold running water. Slice the white and pale green parts and roughly chop. Trim the root portion from the leek and slice off the dark green leafy top. Slice the remaining white and pale green section of the leek lengthwise and carefully rinse, fanning layers under cold running water to wash away any grit trapped inside. Then slice into 1/2 inch pieces.
Cook the soup. Melt butter in a large stock pot or dutch oven over medium flame. Add the green garlic and leek, stirring to coat with butter and cook for 3 to 4 minutes to soften slightly, stirring frequently to avoid browning. Add potatoes and stir to coat with butter. Cook for about a minute, then add broth. Add just enough water to cover potatoes – I added 1-1/2 cups. Raise heat to medium-high and bring to a boil. Reduce heat to low and simmer uncovered until potatoes are cooked through and tender, about 20 minutes. Cool the soup briefly and then refrigerate until completely chilled. You can prepare it up to this point a day a head – in fact, letting it rest in the fridge overnight will bring out the green garlic’s flavor.
Finish the soup. Working in batches, purée the soup. Transfer to a bowl. Stir in the half & half and adjust the seasoning with salt. If the vichyssoise is too thick, stir in a little water, a tablespoon at a time. Don’t overdo the water – you want this soup thick and creamy. Ladle into individual bowls and top with chopped parsley. Serve.
Leftovers. If you’re only going to use part of the soup in a sitting, only add part of the half & half to the portion you’re serving. It will help keep the flavors fresher and creamier when you serve the second portion.