Season each rib generously all over with salt and pepper.
If time permits, set aside for an hour to let the seasoning penetrate the meat.
Pre-heat oven to 350°F.
Put a large cast-iron skillet over medium-high heat.
When pan is hot, add olive oil just to film the bottom.
Working in batches so as not to crowd pan, brown a few short ribs at a time on both meaty sides.
Reduce heat as necessary to achieve browning gradually; it may take 4 to 5 minutes per side for well-browned ribs. This will guarantee a dark, rich color for the sauce.
Transfer ribs to a Dutch oven or deep, wide baking dish. Leave skillet on the heat.
Use a clove to pin a bay leaf to the rounded side of each onion half.
Set the onion cut side down in the skillet and let cut side brown for a minute or two.
Transfer onion to pot with ribs.
Add tomato paste and wine to skillet and bring to a simmer, stirring and scraping with a wooden spoon to dissolve all of the flavorful brown bits, then pour wine mixture over ribs.
Add broth to Dutch oven, cover and bake for 2 to 2 1/2 hours, until meat is very tender when probed.
Remove from heat, uncover and skim fat from surface; see note.
Fill a large saucepan with well-salted water and bring to a boil.
Add carrots and simmer until done, but not too soft, about 8 to 10 minutes. Drain carrots, reserving 1/2 cup of cooking water.
Return saucepan to stove over medium-high heat and add butter.
Add leeks, season with salt, and cook, stirring, until barely softened.
Add carrots and reserved cooking water.
Gently combine, turn off heat and cover for 5 minutes.
Carefully transfer short ribs to a large, deep serving platter.
Bring braising juices to a boil over high heat.
If you wish to thicken the sauce lightly, add potato starch mixture and cook 1 minute more.
Ladle sauce over ribs.
Transfer carrot and leek mixture, along with buttery juices, to a serving dish.
Sprinkle carrots and ribs with parsley and chives and serve.