Position a rack in the center of the oven and preheat to 350 degrees F. Line a standard 12-cup muffin tin with paper liners.
Sift together the flour, baking powder, salt, and cinnamon. Set aside.
In a large bowl, combine the beets, sugar, eggs, oil, and vanilla. Using a rubber spatula, stir in one-third of the flour mixture, and continue stirring just until the flour disappears. Do not beat or overmix. Repeat, adding the remaining flour mixture in 2 batches.
Spoon the batter into the prepared muffin cups, dividing the batter evenly and filling each cup almost to the top of the liner. Bake for 25 to 30 minutes, or until a toothpick inserted into the center of the cupcakes comes out clean. Let the cupcakes rest in the pan, set on a wire rack, for 10 minutes. Transfer the cupcakes to the wire rack to cool completely, about an hour.
To make the buttercream, in the bowl of a stand mixer fitted with the paddle attachment or in a mixing bowl with a handheld electric mixer, cream the butter on low speed. Add the sugar, cream, orange oil, and vanilla, and beat until incorporated, about 2 minutes. Add the orange juice, a little at a time, until the buttercream is fluffy and smooth.
When the cupcakes are completely cool, spread a thick layer of buttercream over the tops, swirling the frosting to decorate the tops. Alternatively, the frosting can be transferred to a pastry bag and piped around the tops of the cupcakes. The cupcakes can be made up to 2 days in advance. Store, covered, at room temperature.