Score the fat on the belly and season both sides with salt and pepper. Make a bed in a roasting pan with the carrots and onions. Lay the pork belly on top, skin-side up. Pour over the cider and a cup of water. Cook 30 minutes to get the fat running. Stir together the honey and Dijon with a spoonful of water. Brush the mustard mixture onto the fat, and lower the heat to 325 degrees F. Continue cooking, basting occasionally with the honey, for 3 hours.
An hour before the belly is done, start the lentils. Rinse the lentils in cold water, drain, and put in a saucepan with 1 1/2 cups water, the shallot, bay leaf and thyme. Bring to a boil, cover, and simmer gently until tender, 30 to 40 minutes, checking occasionally to make sure no more water is needed. Drain the lentils, and season with salt, and pepper. Set aside to reheat later.
Remove the belly from the oven when it's done, and transfer to a baking sheet. Return it to the oven at 450 degrees F for 15 minutes to crispen the skin.
Meanwhile, drain the juices from the roasting pan into a measuring cup, reserving the vegetables and keeping them warm. Pour the fat off the juices, using some to moisten the lentils and saving the rest for another use. Deglaze the roasting pan with the vinegar and reduce to a glaze, then add the cooking juices and cook down for a minute or two. Reheat the lentils gently.
Slice the belly and serve with the vegetables, and with the pork-fat moistened lentils.