Cook butter in a heavy saucepan over medium heat, stirring constantly until it foams, turns clear, and then turns a deep brown, about 6 minutes. Pour browned butter into a large mixing bowl and cool to room temperature. Whisk together brown butter, buttermilk, yeast, salt, eggs, and brown sugar until combined. Add flour and mix in with a wooden spoon or dough whisk until incorporated. Cover bowl with plastic and let rest at room temperature for two hours, then chill in the refrigerator for at least three hours or up to five days.
When ready to bake, line two rimmed baking sheets with parchment paper and flour board or counter well. Use a floured bench scraper to scrape dough out onto board. Dust dough with flour and use a lightly floured rolling pin to roll dough into a 1/2-inch thick rectangle. Use a floured 3-inch cutter to form 14 rounds; cutout the center of each round with a floured 1½-inch cutter. Place doughnuts on prepared baking sheets and let rest at room temperature for 45 minutes.
While the dough rests, make the candied hazelnuts. Preheat oven to 350°F and line a rimmed baking sheet with parchment paper. Whisk together sugar and hot water in a small bowl until sugar is dissolved. Add hazelnuts and turbinado sugar; toss to coat. Spread the hazelnuts in a single layer on prepared baking sheet and bake until crisp, about 10 to 15 minutes. Let cool, then chop with a sharp knife.
Bake doughnuts for 18 minutes, until golden. Place on a wire rack to cool for 5minutes. For the glaze, whisk together melted butter, powdered sugar, salt, Frangelico, and milk until smooth. Set rack with doughnuts over a piece of wax paper. Dip doughnut tops into glaze, sprinkle with candied hazelnuts, and let seton rack before serving.