Preheat the oven to 450°F. Fill a medium pot with water and salt and bring to a boil over high heat. Wash and slice the potatoes into 1/8 inch thick slices. Add the sliced potatoes to the boiling water and cook for 3-5 minutes. Drain into a colander. In a medium bowl, combine egg yolks, milk, salt, nutmeg, and pepper. Whisk to combine and reserve. Lightly coat a pan or baking form with butter. Place the par-cooked potatoes in the pan and add the egg mixture. Top it with shredded cheese. Bake in the oven for 30 minutes until the potatoes are soft and the cheese is brown on top. While the gratin bakes, chop the capers, mince the garlic, and rip the thyme. In a saucepan, melt 4 tablespoons of butter, then add capers, minced garlic, fresh thyme, and lime juice. Let it boil for a minute. Once the potato gratin is done, cut a slice and place it on a plate with the arugula salad over the balsamic glaze. Top the potato gratin with the capers oil and serve.
