Prepare the burgers: Place the beets on a large piece of paper towel. Top with another paper towel and press down firmly to released the excess moisture. Add the beets and remaining burger ingredients to a food processor. Process until the mixture forms a consistency similar to ground meat. Take about ⅓ cup of the mixture and form into a mini patty. Repeat for the rest of the mixture.
Cook the burgers: Heat 1 tablespoon of olive oil in a large grill pan or skillet on the stove over medium-high heat. Once heated, add half of the patties to the pan. Cook on each side for about 3-4 minutes until golden brown. Repeat this step for the remaining patties.
Make the sauce: Combine the Greek yogurt, sour cream, cilantro, and lime juice in a small bowl and stir.
Assemble the sliders: Place one burger in each bun. Top with sprouts and a small dollop of Greek yogurt sauce.