Peel the vegetables then wash them.Slice them thinly using a mandolin. Carefully arrange each vegetable in a baking dish, making sure the slices are vertical. Preheat your oven to 180°C.
Peel and degerm the garlic cloves. Squeeze it over the vegetables.
Carefully pour the vegetable broth over the top of the vegetables using a ladle.
Place a knob of butter on each vegetable.
Sprinkle with parmesan, salt and pepper. Place a few sprigs of thyme on the dish.
Cover your vegetables with a sheet of aluminum foil and bake.Cook for 45 minutes then uncover and continue cooking until your dish is golden brown (5 to 15 minutes depending on whether you want a more or less toasted gratin).