In a large soup pot, melt the butter over low heat, add the leeks and cook, stirring frequently until soft but not browned. Add the turnips, apples, broth, salt and nutmeg. Bring to a boil, turn down to a simmer, and cook for 20-30 minutes until the turnips are very tender.
In batches, puree the soup in a blender until smooth, thinning with more broth or water if you think it’s too thick. Taste, and season with salt and pepper. (At this point, the soup can be refrigerated for 3 days or frozen for longer storage.)
Return to the pot and bring to a simmer. For the balsamic brown butter, melt the butter in a small pot over low heat until browned and nutty. Remove from the heat and swirl in the balsamic vinegar. Ladle the soup into bowls and drizzle with the balsamic butter.