Preheat the oven to 350 degrees. Season the pork liberally with salt and pepper. In a wide pot, heat the oil just until it begins to smoke. Brown the pork on all sides (working in batches if necessary)- you really want to generate nice caramel colored matter on both the pork and the bottom of the pan (but not black! Don’t burn!) Remove the pork, set aside. Drain excess oil.
Add the butter, shallots, and cornichons to pan. Using a wooden spoon, sauté the cornichons and shallots, all the while scraping the bottom of the pan to loosen all brown bits. Add the vinegar and reduce it completely. Then add the wine and reduce by half. Finally, add the chicken stock, thyme, and reserved pork. Bring the liquid to a boil, and put the (uncovered) pot into the oven.
After 20 minutes, start checking the pork for doneness. It should be tender but not dry. When it’s finished, strain out the meat and solids, and reduce the cooking liquid over a medium high flame until thick and unctuous. Add the pork and solids, and season to taste. It can be served right away, but would be even better after sitting in the fridge for a day or two.