1. Make cupcakes: Preheat oven to 350°. Wrap each beet in foil, put on a rimmed baking sheet, and roast until tender when pierced, 1 to 1 1/2 hours. Let cool. Peel, cut into chunks, and purée in a food processor. Measure 1 1/4 cups purée.
2. Line a 12-cup muffin pan (1/2-cup size) with paper liners. Sift together flour, baking powder, salt, and cinnamon in a medium bowl. Set aside.
3. Whisk together beet purée, granulated sugar, eggs, oil, and vanilla in a large bowl. With a plastic spatula, stir in flour mixture one-third at a time just until smooth.
4. Spoon batter into muffin cups, filling each almost to the top of the liner. Bake until cupcakes spring back when gently pressed and a toothpick inserted in center of one comes out with a couple of moist crumbs clinging, 25 to 30 minutes. Let cool in pan on a rack 10 minutes. Transfer to rack to cool completely.
5. Make buttercream: Beat butter with a mixer on high speed until very smooth. Add powdered sugar, orange oil, and vanilla; beat on low speed to blend, then on medium until fluffy and smooth.
6. Spread a thick layer of buttercream on cupcakes, swirling it decoratively. Or pipe frosting from a pastry bag, using a star tip.
Make ahead: Up to 1 day, chilled.
*Find pure orange oil, an essential oil pressed from orange rind (and different from orange extract), at surlatable.com