Preheat your oven to gas 6, 200°C, fan 180°C. Rub a large baking dish with the smashed garlic clove, and then discard the garlic. Put a third of the tallegio slices aside to add to the dish towards the end of cooking.
Make a layer of potato slices in the bottom of the dish, overlapping them slightly. Use the other two thirds of brie and chopped tarragon to sprinkle over the potatoes and continue this process until everything is used up, remembering to season every layer with salt and pepper.
Pour the milk over. The liquid should come up to the bottom of the top layer of potatoes. If it does not, add a little more but be careful not to completely submerge them. Break up the butter into little pieces then dot the top of the potatoes.
Bake for about 1 hour, or until a knife inserted into the top of the gratin doesn’t meet with any resistance. Remove from the oven, place the remainder of the cheese slices on top and return it to the oven for another 5-10 minutes, or until the cheese is melted and turning golden.
Allow the dish to stand for about 5 minutes, and then serve it with a few fresh tarragon leaves on top, if desired.