Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes. Drain. Cool until easily handled. Peel potatoes and mash in a bowl.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler. Grease a baking dish generously with 1 tablespoon butter.
Heat milk in a saucepan over medium heat, add remaining 1 tablespoon butter, and melt. Season with salt and mix into mashed potatoes. Whisk in egg yolks, chives, nutmeg, and pepper briefly. Spoon mashed potatoes into the prepared baking dish and cover with mozzarella slices.
Bake in the preheated oven until cheese is melted and lightly browned, about 10 minutes.