Heat the oven to 400° F.
Place 3 of the beets on a large sheet of foil and wrap tightly. Repeat with the remaining beets. Roast until tender, about 1 hour. Let cool for about 10 minutes. Using a small knife, peel each beet and trim the ends. Place a piece of parchment or plastic wrap on a cutting board. Slice the beets 1/8 to 1/4 inch thick; set aside to cool completely.
Meanwhile, in a small bowl combine the goat cheese, chives, salt, and pepper.
To assemble, place 1 beet slice on a platter. Spread 1/2 teaspoon of the cheese mixture over the beet. Top with another slice; repeat with one more slice of beet and 1 more teaspoon of the cheese mixture. You should have a stack of three slices of beet with some cheese mixture in between each slice. Repeat with the remaining beets and cheese mixture. Arrange the napoleons on a platter and drizzle with the chive puree. Serve immediately.
Tip: To check a beet for doneness, insert the tip of a knife or the tines of a fork into the center. If there is no resistance, the beet is done.