Preheat the oven to 350ºF. Grease the inside of an 8 × 8-inch baking dish with butter.
Wash and peel the potatoes, reserving the peels. Set aside.
Wash the squash. Remove and discard 1 inch off the top and the bottom. Using a vegetable peeler, and working away from yourself, peel the squash until you reach the deep-orange flesh. Remove and discard the seeds, but reserve the peels. Set aside.
Fill a large bowl with cold water and set nearby. Using a mandoline slicer, slice the potatoes and squash paper thin, then transfer the slices to the bowl of water.
In a large pot, heat the cream, honey, soy sauce, and reserved squash and potato peels over medium heat and cook until the liquid has reduced by ⅓, about 30 minutes. Add the butter. Pour the mixture into a blender and blend until smooth.
In a bowl large enough to hold the cream mixture, place the egg yolks. Whisk the yolks continuously while slowly pouring the warm liquid into the bowl. The goal is to temper, not cook, the eggs.
In the prepared baking dish, place the potatoes and squash in alternating layers (potato, then squash, then potato, and so on), slightly overlapping each layer. Once a layer is finished, add just enough of the cream mixture to cover the vegetables. Repeat this process until all of the vegetables are used. Place the baking dish on top of a baking sheet to catch drips. Cover the baking dish with aluminum foil and bake for about 45 minutes to 1 hour. Remove from the oven, remove the foil, and top with the parmesan cheese. Change the oven setting to broil. Cook under the broiler for another 5 to 7 minutes, until the cheese is browned on top.
Remove from the oven. Let the gratin rest for about 15 minutes before serving.