Rich Beef Broth with Carrot

Put oven rack in lower third of oven and preheat oven to 450°F.
Spread short ribs, shanks, onions, celery, and quartered carrot in a large flameproof roasting pan and roast, turning occasionally, until well browned, about 1 hour.
Wrap parsley, thyme, bay leaf, and peppercorns in cheesecloth and tie into a bundle with string to make a bouquet garni.
Transfer meat (with bones) and roasted vegetables to a 6- to 8-quart pot and add 12 cups water, salt, and bouquet garni. Straddle roasting pan across 2 burners, then add remaining 2 cups water and deglaze pan by boiling, stirring and scraping up brown bits, 1 minute. Add deglazing liquid to pot, then bring to a boil, skimming froth. Reduce heat to low and gently simmer, uncovered, until liquid is reduced to about 5 cups, 7 to 8 hours.
Transfer mixture to a bowl and cool completely, uncovered, about 1 hour, then chill, covered, at least 8 hours.
Remove fat. Reheat mixture in cleaned pot over moderate heat until warm. Discard beef shanks (meat and bone), then transfer short ribs to a platter and, when cool enough to handle, discard bones and fat. Finely chop short ribs and reserve for terrine . If not using meat within 1 hour, chill, covered. Pour broth though a cheesecloth-lined sieve into a 1 1/2-quart saucepan and skim off any remaining fat. Bring broth to a simmer and season with salt.
Cook diced carrot in a small saucepan of boiling salted water until just tender, about 4 minutes. Drain in sieve and divide among demitasse cups, then add broth.