Preheat oven to 300°. Prepare ribs: In a large roasting pan, combine all ingredients except for ribs. Set ribs in pan and rub all over with achiote mixture. Seal pan tightly with foil. Roast, turning ribs every hour, until meat is tender when pierced but not falling off bones, about 2 hours.
Meanwhile, make sauce: In a medium saucepan, combine all ingredients except broth and liquid smoke. Bring to a simmer over medium heat and cook, stirring often, until reduced to a thick paste, 18 to 20 minutes. Remove from heat.
Transfer ribs to a rimmed baking sheet and set aside. Pour pan juices into a large glass measuring cup; skim and discard fat. Add chicken broth to make 2 1/2 cups, then pour into pan with sauce paste. Bring to a simmer over medium heat and cook, stirring often, until ancho pieces are very soft, 15 to 20 minutes. Meanwhile, prepare a grill for medium (350° to 450°) indirect heat, but skip the drip pan.
Discard cinnamon stick from sauce and stir in liquid smoke. Purée sauce in a blender, then strain. Add more lime juice and salt to taste. Pour 1 cup sauce into a small bowl for basting and the rest into a small pitcher.
Combine ingredients for relish with salt to taste in a medium bowl; set aside.
Grill ribs, covered, over direct heat, turning once, until browned and sizzling, 4 to 8 minutes total. Transfer ribs bony side up to indirect-heat area and brush generously with some of the 1 cup sauce. Cook until sauce is set, about 5 minutes, then turn, brush again, and cook a few more minutes. Transfer short ribs to plates or cut spareribs between bones and put on a platter. Serve ribs with reserved sauce and the avocado relish.
*Find achiote paste, a Yucatecan seasoning blend made with annatto and vinegar, in the Latino-foods aisle. Ask a butcher to cut short ribs in half.
Secrets for Fantastic Ribs:
USE PORK SPARERIBS AND BEEF SHORT RIBS. More generously marbled than pork baby back ribs, country-style spareribs, or regular beef ribs, they're noticeably more juicy and flavorful.
TRIM SPARERIBS ST. LOUIS-STYLE. Meaning, trim them into a tidy, rectangular shape that cooks evenly. Ask a butcher to trim them, or do it yourself: On the bony side, trim the flap of meat from the center, flush with the bones. Then cut the rack lengthwise between the 4- to 5-in.-wide rib section and the chewy skirt (above). Save scraps for soup.
REMOVE THE MEMBRANE. Pork spareribs have a membrane on the underside that can shrink up and make the meat cook unevenly. It's easy to remove: Slide the tip of a meat thermometer under the membrane at one end to loosen an edge. Pull off membrane with a paper towel (it may come off in pieces) while holding the rack down with your other hand.
Setting Up Your Grill:
MASTER THE ART OF INDIRECT HEAT. This allows the ribs to cook long and slow, so they become extremely tender and rendered of much of their fat. Then brown them over the direct-heat part of the fire.
If using gas, put a drip pan in place under one part of the cooking grate (the indirect-heat area), then light only the burner or burners on the rest of the grill (the direct-heat area).
If using charcoal, ignite 50 briquets in a chimney, then bank coals on opposite sides of firegrate, leaving a cleared area in the middle. Set a drip pan in the cleared area. Let coals burn down to the temperature specified in the recipe. To maintain the temp during cooking, add 5 briquets to each mound of coals about every 30 minutes, starting when ribs go on grill; if fire gets too hot, partially close vents under grill and on lid.