Divide ribs between 2 large pots. Cover with cold water. Bring to a boil. Reduce to a simmer, and cook 3 minutes, skimming any foam that rises to the surface. Drain; rinse under cold water. Cover ribs with cold water again.
Divide celery, carrots, onion, and garlic between pots; add 1 tablespoon salt to each pot. Bring to a boil. Reduce to a simmer, and cook until ribs are tender, about 1 hour. Drain; discard vegetables.
Preheat grill to medium (if you are using a charcoal grill, coals are ready when you can hold your hand 5 inches above grill for just 5 to 6 seconds). Season ribs generously with salt and pepper; brush with barbecue sauce. Grill, turning frequently and brushing with sauce occasionally, until heated through and marked by grill, about 15 minutes. Cut into servings of about 3 ribs each; brush with more sauce. Serve immediately.