Rinse pork and pat dry. Score skin in a crisscross diamond pattern, making 1/8-inch-deep cuts about 1 inch apart. Separate and peel garlic cloves. In a mortar and pestle, crush garlic, salt, oregano, thyme, and pepper into a coarse paste (or mince garlic, then mix with salt, herbs, and pepper). Rub garlic paste all over roast. Set roast, skin side up, on a rack in an oiled 9- by 13-inch roasting pan.
Roast in a 450° oven (see notes) until deep golden brown, 40 to 45 minutes.
Remove pan from oven and scatter onion and carrot slices around pork. Pour 1/4 cup sherry and 2 cups broth into pan. Add chile flakes. Baste pork with some of the pan juices. Reduce oven temperature to 225° and bake until a thermometer inserted through the center of thickest part at bone reads 170° to 175°, 8 to 9 hours (do not insert thermometer before the last hour of roasting). If pork is done before you're ready to serve, reduce oven temperature to 160° and hold in oven up to 4 hours.
About 30 minutes before serving, transfer pork to a carving board. (For extra-crisp pork skin cracklings, see notes.) Cover pork loosely and set in a warm place. Pour the remaining 1/4 cup sherry and 2 cups broth into the roasting pan (drippings will be dark) and set on a burner over high heat. Bring to a boil, scraping up browned bits from bottom of pan. Boil, stirring occasionally, until liquid is reduced by about half, 7 to 10 minutes. Pour through a wire strainer set over a bowl, pressing on vegetables (discard vegetables); you should have 1 to 1 1/3 cups drippings. Skim off and discard fat. Stir in vinegar.
Lift skin off pork and cut it into bite-size chunks or strips; pile on a platter. Slice meat across the grain and arrange on platter. Drizzle about a fourth of the pan juices over meat; serve remaining to add to taste.
NOTES: You may need to order the pork from your meat market in advance. Monitor oven temperature with a thermometer before starting, and adjust temperature as needed. If you hold the roast in the oven after it is done, check to see if oven is still on after 12 hours of continuous use; some ovens turn off automatically. A convection oven is not recommended. For extra-crisp cracklings, pull skin off cooked pork, separate into bite-size pieces, and place in a shallow pan. Bake in a 400° oven until crisp and puffy, about 10 minutes.