Preheat the oven to 350° F (176°C).
Roast the beets on a baking sheet in the oven until tender. If using baby beets, this should take 20 to 30 minutes (or longer, if the beets aren't fresh); if using regular-size beets, this could take up to an hour. Place the roasted beets on a towel and gently rub to remove their skins. Let cool to room temperature. Toast the pine nuts on a second baking sheet for 8 minutes and transfer to a plate to cool.
After the beets have cooled, slice off the top and bottom of each beet. If using baby beets, cut each in half. If using regular-size beets, cut into small wedges. Season with salt and pepper.
Slice each tangerine into thick rounds, then slice each round in half crosswise to form a sort of tangerine half moon.
Arrange 5 beet pieces and 5 tangerine slices on each plate. Drizzle each with 1/2 teaspoon lemon juice and 1 teaspoon olive oil on top and sprinkle with 1/2 teaspoon shallot, 1 teaspoon of the toasted pine nuts, and 1/2 teaspoon basil. Season with salt and pepper.