Place 1/2 cup orange juice, orange zest and vanilla bean pieces in a small saucepan and bring to a boil. Remove from heat and set aside.
Reheat orange-vanilla mixture. Dissolve cornstarch in remaining 1/4 cup orange juice. Add to orange-vanilla mixture, stirring constantly. Cook until thickened and clear. Add cardamom, salt and pepper. (Add vanilla extract if not using vanilla bean.) Remove pieces of vanilla bean, scraping seeds into mixture. If glaze is too thick, add more orange juice to thin it.
Pour over beets, glazing beets thoroughly. Reheat, if necessary, before serving.