Step 1: In a small bowl, combine the warm milk, 1 teaspoon of granulated sugar, and active dry yeast. Stir gently and let sit for 5-10 minutes until foamy, indicating the yeast is active.
Step 2: In a large bowl, whisk together the remaining ¼ cup granulated sugar, egg, melted butter, and salt. Pour in the yeast mixture and stir to combine.
Step 3: In a separate bowl, whisk together the all-purpose flour and cocoa powder. Gradually add the dry ingredients to the wet ingredients, mixing until a shaggy dough forms. Turn the dough out onto a lightly floured surface.
Step 4: Knead the dough for 5-7 minutes until it is smooth and elastic. If it's too sticky, add a tiny bit more flour, but be careful not to over-flour. Lightly grease a clean bowl, place the dough in it, turning once to coat. Cover with plastic wrap or a damp towel and let rise in a warm place for 1-1.5 hours, or until doubled in size.
Step 5: Once risen, gently punch down the dough. Roll it out on a lightly floured surface to about ½-inch thickness. Use a 3-inch doughnut cutter (or two different sized round cutters) to cut out doughnuts and their holes. Gather and re-roll scraps once.
Step 6: Place the cut doughnuts and holes on parchment-lined baking sheets, leaving some space between them. Cover loosely with plastic wrap or a light towel and let rise for another 30-45 minutes, or until noticeably puffy.
Step 7: While the doughnuts are rising for the second time, prepare the glaze. In a medium bowl, whisk together the powdered sugar, milk/cream, melted butter, chocolate hazelnut spread, and vanilla extract until smooth. Adjust milk as needed for desired consistency.
Step 8: Heat 3-4 inches of vegetable oil in a large heavy-bottomed pot or Dutch oven to 350-375°F (175-190°C). Use a thermometer to monitor the temperature.
Step 9: Carefully place 2-3 doughnuts (don't overcrowd the pot) into the hot oil. Fry for 1-2 minutes per side, until golden brown and cooked through. Remove with a slotted spoon and place on a wire rack set over paper towels to drain excess oil. Repeat with remaining doughnuts and holes.
Step 10: While the doughnuts are still warm, dip each one into the chocolate hazelnut glaze, allowing excess to drip off. Place back on the wire rack. Sprinkle immediately with toasted, chopped hazelnuts. Serve warm and enjoy!