1In a large Dutch oven over medium-high heat, sear the pieces of pork belly until golden brown (about 1-2 minutes per side). Transfer the meat to a plate and drain off all but 1 Tablespoon of fat from the pan.
2Lower heat to medium and add the oinonand shallots. . Cook, stirring frequently until tender and beginning to caramelize, about 8-10 minutes. Stir in the garlic and cook 1-2 minutes more.
3Add the beer, brown sugar 1 teaspoon salt and the pepper. Bring to a boil and reduce liquid by half, about 10 minutes.
4Return the ieces of pork belly to the pot and add the carrots and broth. Bring up to a simmer and partially cover the pan. Cook 1 hour 45 minutes to 2 hours or until the meat is meltingly tender.
5With a slotted spoon, transfer the pork belly to a warm plate, cover with foil and reserve.
6Bring the remainng ingredients in the pot up to a boil and reduce to 2-1/2 Cups liqud. Working in batches as needed, transfer the mixture to a blender and carefully purre, reserving some of the whole carrot pieces. Return the puree to the pan then stir in the heavy cream, and vinegar if used.
7Taste and add salt and or pepper as needed. Serve with the pork belly in bowls, as is or over mashed potatoes, pasta or rice.