Season the meat with salt and cayenne. In a pot large enough to hold the meat, heat the oil over a high flame, add the meat, and brown on all sides. Remove the meat to a plate.

Meanwhile, chop the carrots, celery, onion, and jalapenos, and mince half the garlic. After removing the meat, add the butter to the the pot and melt. Add the vegetables, season with salt and cayenne, and cook until they begin to soften. Return the meat to the pot and add the thyme, bay leaves, stocks, and enough water nearly to cover the meat. Bring to a boil, cover the pot, reduce the heat, and simmer for at least three hours, until the meat crumbles under a hard stare.

While the meat is cooking, prepare the green beans and mayonnaise:

Trim the ends from the beans, cut in half lengthwise along the seams (French cut), and cut into two-inch lengths. Cook the beans in boiling water for several minutes, until just tender.

Mince the remaining garlic with a pinch of salt. Blend the egg yolk, garlic, lemon juice, and mustard in a food processor with a steel blade. With the motor running, add the oil in a thin stream. The result should be a fairly stiff mayonnaise. Refrigerate until the sandwiches are served.

When the meat is done, remove it to a platter and discard any bones along with the thyme and bay leaves. Slice the meat with a sharp knife. (If the meat is properly cooked, this will be a frustrating process.) Remove the debris (crumbs and small chunks), which should amount to a cup or more. Cover the rest of the meat and keep warm.

Skim the fat from the cooking liquid, return the debris to the pot, and reduce over high heat until thick enough to coat the back of a spoon.

Cut each roll in half and spread mayonnaise on both sides. Make the sandwiches with generous portions of meat covered with sauce and a layer of green beans.