Preheat the oven to 350 degrees. Trim the leafy tops, including stems, from the beets, rinse thoroughly, and set aside. Arrange the beets in a roasting pan just large enough to hold them in a single layer. Drizzle the beets with 1 tablespoon of the olive oil and cover the pan with aluminum foil.
Bake the beets until tender when pierced with a knife, about 50 minutes. Set aside until cool enough to handle.
While the beets are cooking, roughly chop the reserved beet tops, including the stems. Heat 1 tablespoon of the olive oil in a large pot over medium-low heat. Add the chopped beet tops and stems and pour in ¾ cup water. Cover and cook, stirring often, until they are wilted and tender, about 30 minutes.
Peel the beets, cut into large chunks, and add them to the beet tops in the pot. Stir in the tarragon and salt.
Turn the mixture into a serving dish and drizzle with the remaining 2 tablespoons oil.