Preheat the oven to 425F degrees.
Trim the greens from the beets, leaving about 1-inch of the stems. Wash and drain beets in a colander. Wrap beets in aluminum foil "bundles" and place on a rimmed baking sheet. Cook in oven for about one hour or until the beets are tender enough to be pierced with a fork. Remove the tray from the oven, open the foil packets and let the beets cool on the tray until cold enough to handle. Cut off the top and bottom of beet and rub or pare off the skins. Cut the beets in slices.
In a medium bowl, whisk together the grapefruit juice, vinegar, and maple syrup or honey.
Place the cornstarch in a small saucepan and whisk in the grapefruit mixture until all the cornstarch is dissolved.
Heat the pan on medium heat until just boiling, whisking frequently. Turn the heat to medium-low and cook, stirring often, for about 5-8 minutes, or until the glaze is thickened and glossy. Remove from the heat.
Drizzle the hot glaze over beets right away if you want to serve as a hot dish. If you are serving cold, drizzle the glaze on the beets and then refrigerate at least one hour and up to 24 hours ahead of serving.