Position a rack in the middle of the oven and heat the oven to 375°F.
Coat the inside of a 9x9-inch or equivalent baking dish with 1 tbsp. of the olive oil.
Peel the potatoes, slice them into 1/8-inch-thick rounds, and put them in a bowl.
Add the thyme, 1½ tsp. salt, and a few grinds of pepper and toss to coat everything evenly, making sure that you separate all the sticking potatoes.
Pour the potatoes into the baking dish and arrange them in an even layer.
Bring the chicken broth to a simmer in a small saucepan.
Whisk the remaining 4 tbsp. oil into the broth and pour the mixture over the potatoes.
Press down on the potatoes with the back of a spatula to distribute the liquid.
Cover the baking dish with foil and bake for 30 min.
Remove from the oven and remove the foil.
Press down on the potatoes with a spatula to get the juices to bubble up over the edges of the potatoes.
Return the pan to the oven, uncovered now, and cook for another 15 min.
Repeat this process of pressing on the potatoes every 15 min. two more times for a total cooking time of 1 hour and 15 min.
When done, the gratin will be lightly brown on top and the potatoes tender when pierced with the tip of a small knife. If the top of the gratin isn’t browned after this amount of time, press on the potatoes again with the spatula and return the pan to the oven for up to another 10 min.
Remove the dish from the oven.
Press one last time with the spatula and then let the dish rest on a cooling rack for 30 min. before serving.