Preheat the oven to 400 F (200 C).
Rub the beef shanks with oil, season with salt and pepper, and place them in a baking dish with high sides.
Peel the celery, removing the filaments from the outer stalks, then cut them in half lengthwise and slice them thinly.
Remove the inner leaves and set them aside in the refrigerator with the tarragon leaves.
Arrange the celery, shallots, and garlic around the shanks and put the dish in the oven.
After 40 minutes, when the shank is browned, remove it from the oven.
Lower the oven temperature to 350 F (180 C).
Remove the shanks from the dish and remove excess fat with a spoon.
Return the shanks, pour the dark beer over them, and add the tomato paste.
Cover with a lid or a sheet of aluminum foil and bake for 4 hours or until the meat is tender and falls off the bone.
Check halfway through cooking and if the liquid has reduced too much, add water.
Remove the dish from the oven and uncover it.
Baste the shanks with the pan juices, then return to the oven for an additional 10 to 15 minutes to reduce the juices and caramelize the meat.
Garnish with celery and tarragon leaves and serve with mashed potatoes.