Preheat the oven to 375 degrees F. Butter the inside of a 9-inch spring form pan with 1 teaspoon of the butter. Sprinkle in 2 tablespoons of the breadcrumbs and rotate the pan to coat the bottom and the sides completely. Set aside.
Put the potatoes and garlic in a large saucepan and cover with cold water. Season the water well with salt and 2 tablespoons butter and bring to a boil over high heat. Simmer until cooked through and easily pierced with the tip of knife, about 15 minutes. Drain well.
In a small saucepan over medium heat, melt the remaining stick of butter. In a large bowl combine the melted butter, sour cream, 1 cup Parmigiano Reggiano and 3/4 teaspoon salt. Mix well with a rubber spatula.
Using a ricer, rice the cooked potatoes over the sour cream mixture. Use the rubber spatula to combine well. Add the beaten eggs and stir to combine. Scoop half of the mixture into the prepared pan. Sprinkle the finocchiona evenly over potato mixture and top with the provolone. Scoop the remaining potato mixture on top and spread evenly. Sprinkle with the remaining 2 tablespoons bread crumbs, 1/2 cup Parmigiano Reggiano and top with the remaining 5 teaspoons of butter. Bake until hot all the way through and golden brown on top, about 45 minutes. Allow the gateau to rest for 15 minutes before unmolding and serving.