Poke beets all over with a fork; transfer to a microwave-safe bowl. Cover bowl with plastic wrap, leaving some room for air to escape. Cook beets in the microwave for 4 minutes. Wrap each beet with aluminum foil and place on a baking sheet.
Season garlic halves with 1 tablespoon olive oil, salt, and pepper. Wrap each half with aluminum foil and place on the baking sheet.
Bake in the preheated oven until beets are tender when pierced with a fork, about 35 minutes.
Cool beets until running water for 1 minute. Peel off skins and transfer to a food processor.
Squeeze garlic cloves into the food processor. Drizzle in any olive oil remaining on the aluminum foil. Add chickpeas, 1/4 cup olive oil, lemon juice, tahini, cumin, salt, pepper, and cayenne. Pulse until combined. Add Greek yogurt; process until smooth.