Preheat the oven to 350 degrees F. Line two 12-cup muffin tins with paper liners. Set aside.
Wrap the beets in parchment paper. Microwave on high until they are tender (about 8 to 10 minutes). Allow the beets to cool, still wrapped in the paper, until they are just warm to the touch.
Coarsely chop the beets. Add them, along with the lemon juice, to a food processor. Process until they are finely chopped. Stop and scrape down the sides of the bowl as needed.
Add the oil and buttermilk and process until smooth. Add the eggs and process until completely combined.
In a large bowl, whisk together the flour, sugar, cocoa, baking powder, salt, and baking soda. Add the beet mixture, and whisk until just combined.
Using an ice-cream scoop, fill the prepared cupcake liners two-thirds full with batter.
Bake 15-18 minutes, or until a toothpick inserted in center comes out clean. Allow the cupcakes to cool in the pans on wire racks for about 5 minutes. Remove the cupcakes from the pans to the racks, and allow them to cool completely.
Combine the softened cream cheese and butter in the bowl of a stand mixer. Beat until creamy and well-combined.
Add the vanilla extract and kosher salt. Beat again until combined.
Gradually add the powdered sugar on low speed until it is completely combined. Beat on medium-high speed for a few minutes to really whip things up.
Frost the cooled cupcakes.
Notes: Do not use dark cocoa powder. It will really mask the red color that the beets impart.