Step 1At least 2 hours and preferably the night before, combine the Fontina in a glass bowl with 1 cup of the milk. Let stand (refrigerate if standing more than 2 hours).
Step 2Peel the potatoes and cut them into chunks. Place in a large pot and cover with cold water. Add about 1 teaspoon salt, cover and bring to a boil. Uncover, then cook until very soft but not mushy, about 20 to 25 minutes. Drain well, then return to the pot. Mash in 4 tablespoons of the butter and the remaining one-half cup milk. Season with one-half teaspoon salt and one-fourth teaspoon pepper or to taste. Spoon into a large bowl and set aside to cool.
Step 3Heat 4 tablespoons of the butter in a large nonstick skillet over medium heat and add the garlic. Cook 1 to 2 minutes, until aromatic, then add the shiitakes and cook, stirring often, for 10 minutes. Stir in the tamari. Add to the potatoes and mix well. Season liberally with 1 teaspoon salt and one-fourth teaspoon pepper or to taste. Cool slightly, then, using a rubber spatula, fold in the lightly beaten whole eggs until thoroughly blended.
Step 4Use 1 tablespoon of the butter to thickly grease a 9-by-3-inch springform pan. Dust with 2 tablespoons of the panko to coat the entire interior, shaking out the excess. Spoon the potatoes in the pan, smoothing the top. Sprinkle with enough of the remaining panko to barely cover the top. Dot with 1 tablespoon butter. Bake until firm, about 1 hour.
Step 5About 15 minutes before the potatoes are done, melt the remaining 4 tablespoons butter in the top of a double boiler or a stainless steel bowl set over gently simmering water in a regular pot. When the butter is completely melted, stir in the Fontina and milk. Whisking constantly, add the egg yolks one at a time. Continue whisking constantly until the cheese is completely melted and the sauce is glossy and thickened to the consistency of hollandaise, 10 to 15 minutes. Remove from the heat as soon as it is done and transfer to a warm bowl.
Step 6Let the baked timballo stand 10 minutes before running a spatula around the perimeter to unmold. Spoon a generous amount of the fonduta onto small serving plates and top with wedges of the timballo. Serve immediately.