Preheat an oven to 375 F. Grease a 2 qt ceramic baking dish with the softened butter.
Grate the cheese. Grate all three cheeses on a fine grate on a box grater and mix them together well. Set aside 1 cup of shredded cheese.
Mix the sauce. In a large mixing bowl, whisk together the cream, dijon, garlic, chopped herbs, nutmeg, salt, and pepper until smooth. Fold in all but the reserved cup of shredded cheese.
Slice the potatoes. Rinse and peel the potatoes. Use a mandoline to evenly slice the potatoes 1/8" thick (about 2.5-3 mm). Add them to the cream mixture and toss really well, making sure to separate any potatoes that are stuck together so they're coated in cheese and cream.
Layer the gratin. Grab small stacks of potatoes and arrange them in the prepared baking dish edge side up. If you're using an oval dish, line the edges of the dish first and then the middle. If using a rectangle baking dish, make 3 lines of potatoes in the dish. Pour the rest of the cream from the mixing bowl over the potatoes, moving them around so the cream seeps in.
Bake. Line a baking sheet with foil or parchment and then place the baking dish on top. This helps avoid a catastrophe in case there is a spill-over. Cover the baking dish with foil and bake for 40 minutes. Remove the foil and bake for another 30 minutes. Top the gratin with the reserved cheese and bake another 30 minutes. Let the potatoes rest for 20-30 minutes before serving.