Place the beets in a medium saucepan and cover with cold water (water be about 2 inches higher than the beets). Bring to a simmer and cook until the beets are tender when pierced with a fork, about 25 minutes.
Remove the beets from saucepan, let cool slightly, then peel. Cut the beets into ½-inch dice and transfer to a bowl.
In a small bowl, whisk together the vinegar, olive oil and agave nectar. Pour over the beets and stir to combine.
Fill each fillo shell with ½ teaspoon goat cheese. Gently press the goat cheese into bottom of each shell.
Divide the beet mixture evenly between the filled fillo shells. Sprinkle the chopped pistachios over the beets. Serve.