Rub the rack of lamb with a little olive oil and distribute half of the thyme on the previously well-pared meat. Wrap the meat tightly in cling film. Place in the refrigerator for 24 hours, then take out early before searing so that the meat reaches room temperature.
Remove the thyme from the meat, season with salt, pepper and fry well in olive oil with the remaining half of the thyme bunch and 3 cloves of garlic, so that the thyme lies between the meat and the bottom of the pan. Make sure that this bunch of thyme does not have too dense branches so that the meat is seared properly.
Then place the square on a flat tin lined with aluminum foil and place in the 150 ° warm oven for 15 minutes. Turn there several times.
For the sauce, sauté the finely chopped shallots and garlic cloves in a little olive oil, pour in the wine, stock and balsamic vinegar and slowly reduce to 100 ml. Mix very carefully or, depending on your taste, pass through a sieve. Season this amount with salt and pepper and reduce again to 50 ml.
Then bind this reduction, then no longer boiling, with the cold butter piece by piece.
Wrap the meat in aluminum foil and let it rest on the open oven door for about 10 minutes, then cut it open and serve with the sauce.
Long green beans and / or potato gratin go well with it.