Remove short ribs from refrigerator 30 minutes before preparing. Preheat oven to 325°F. Pat short ribs dry using paper towels, and sprinkle evenly with salt and pepper. Heat oil in a large Dutch oven over medium-high. Working in 2 batches, if necessary, add short ribs, and cook until browned on all sides, about 4 minutes per side. Transfer to a large plate or bowl, reserving drippings in Dutch oven; set short ribs aside.
Add onion, carrots, and celery to drippings in Dutch oven; cook over medium, stirring occasionally, until vegetables start to soften, about 5 minutes. Add garlic and tomato paste; cook, stirring constantly, until fragrant and tomato paste evenly coats vegetables, 1 to 2 minutes. Add red wine; bring to a boil over medium-high. Boil, stirring occasionally, until mixture starts to thicken slightly and the smell of alcohol from wine is still present but no longer very pungent, 2 to 3 minutes. Add cola and beef stock; bring to a boil over medium-high. Reduce heat to medium-low, and nestle short ribs back into Dutch oven. Tie together thyme and parsley sprigs using kitchen twine, and nestle into cola mixture; add bay leaves. Cover and bake in preheated oven for 2 hours.
Uncover Dutch oven, and continue cooking, undisturbed, until sauce starts to thicken and reduce and short ribs are very tender when pierced using a fork, about 45 minutes. Remove from oven, and skim fat from top; discard fat. (At this point, short ribs and cola mixture may be refrigerated overnight. Once refrigerated, spoon off solidified fat from top, and discard.)
Transfer short ribs to a large bowl. Place Dutch oven over medium heat; cook, stirring occasionally, until liquid is thickened and reduced enough to coat the back of a spoon, about 15 minutes. Remove from heat, and stir in butter and sherry vinegar. Season with salt to taste. Return short ribs to Dutch oven; cover and keep warm. (To reheat, place Dutch oven over low; heat, covered, 5 minutes before risotto is finished cooking.)
Bring vegetable broth to a boil in a medium saucepan over medium-high. Reduce heat to medium, and maintain a simmer. Meanwhile, heat 2 tablespoons of the butter in a large skillet over medium-high. Place remaining butter back in refrigerator. Add shallots to melted butter in skillet, and cook, stirring occasionally, until translucent, about 5 minutes. Add rice, and cook, stirring constantly, until rice is hot and well coated in butter, about 2 minutes. Add wine and salt, and cook until wine is almost completely evaporated and absorbed, about 1 minute. Ladle about 1 cup simmering broth over rice until just covered. Cook, stirring occasionally, until rice absorbs most of broth. Continue adding broth, 1/2 cup to 3/4 cup at a time (about 8 to 10 times), stirring occasionally, until nearly absorbed between additions; cook until al dente, about 16 minutes total. Once rice is al dente, add about 3/4 cup broth (you may not use all of broth; save remaining broth for another use), and continue cooking until just past al dente (rice will be tender but still have some slight resistance), 1 minute, 30 seconds. Immediately remove from heat, and add Parmigiano-Reggiano and remaining 6 tablespoons cold butter. Stir until butter is completely melted and risotto is thick and smooth but still slightly runny. Transfer to a large bowl or individual plates, and top evenly with short ribs and desired amount of cola sauce. Garnish with thyme leaves; serve immediately.
