The attractively exotic blend of spices that Yucatecan cooks combine with tangy vinegar and roasted garlic is a most satisfying creation. I like to use bone-in, skin-on chicken in this dish for wonderful flavor and moist texture. —Rick Bayless
Put garlic, oil, vinegar, oregano, pepper, cinnamon, sugar, cloves and ½ teaspoon salt into a blender. Process until smooth.
Scrape half of the spice mixture into a small saucepan. Add the broth and ¼ teaspoon of salt, depending on the saltiness of the broth. Set aside.
Put chicken into bowl. Scrape the remaining spice mixture over the chicken and spread it with a spoon to coat chicken evenly. Let stand while you prepare the beets and the grill.
Put sliced beets into a microwave-safe bowl, toss with a sprinkling of salt and 2 tablespoons water. Cover bowl with plastic wrap. Poke a couple of holes in top and microwave on high, stirring once or twice, until tender, about 4 minutes. Pour off all the water and season again with salt.
Heat one side of a gas grill to medium, the other side to medium-low. Or light a charcoal fire and let it burn until the charcoal is covered with white ash (and about medium hot); bank the coals to one side.
Put the seasoned chicken pieces on the grill over the medium-hot fire, skin-side down. When richly browned, usually 5 or 6 minutes, use a pair of tongs to flip them over onto the cooler side of the grill. Cook until completely tender at the bone (the juices will run clear when a fork pierces the thickest part)—expect about 10 minutes longer, maybe more if the breasts are large. Remove from grill to a serving platter.
Bring the broth mixture to a boil; taste and season with salt. Arrange beets around chicken. Douse everything with the hot broth mixture. Sprinkle with cilantro. Pass the salsa.