Combine warm milk, 1 tablespoon of granulated sugar, and active dry yeast in a small bowl; stir and let sit for 5-10 minutes until foamy to ensure yeast activation.

In a separate medium bowl, whisk together the eggs, the remaining ¼ cup of granulated sugar, and vanilla extract; pour this into the foamy yeast mixture and whisk well.

Whisk together the all-purpose flour and salt in a large bowl or the bowl of a stand mixer.

Gradually pour the wet ingredient mixture into the dry ingredients, mixing on low speed or by hand until a shaggy dough forms with no dry spots of flour remaining.

Knead the dough for 5-10 minutes until somewhat smooth and elastic; slowly add the softened unsalted butter, one tablespoon at a time, and continue kneading for another 5-7 minutes until the dough is very smooth, soft, and elastic.

Transfer the dough to a lightly greased large bowl, turning once to coat, then cover with plastic wrap or a kitchen towel and let rise in a warm spot for 1.5-2 hours until doubled in size.

Gently punch down the risen dough to deflate it, re-cover, and refrigerate for at least 4 hours or preferably overnight, as chilling makes the dough easier to handle and improves flavor.

On a lightly floured surface, roll the cold brioche dough until it is about ½-inch thick.

Use a 3-inch donut cutter or two different-sized round cutters to cut out donut shapes and holes; re-roll any scraps gently to cut additional donuts if desired.

Carefully transfer the cut donuts and donut holes to baking sheets lined with parchment paper, ensuring at least 2 inches of space between each.

Loosely cover the baking sheets with plastic wrap or a clean kitchen towel and let them rise in a warm place for 30-45 minutes until visibly puffy and almost doubled in size.

Preheat your oven to 375°F (190°C) about 10-15 minutes before the second rise is complete.

Bake the donuts for 8-12 minutes until golden brown and puffy; donut holes will bake faster, typically 5-7 minutes, so watch them carefully.

Remove the baking sheets from the oven and transfer the baked donuts and holes to a wire rack to cool slightly for about 5 minutes.

While the donuts are cooling, whisk together the ½ cup granulated sugar and 1 tablespoon ground cinnamon in a shallow dish, and melt the ¼ cup unsalted butter in a microwave-safe bowl.

While the donuts are still warm, brush each donut and hole liberally with the melted unsalted butter, then immediately and generously roll each in the cinnamon sugar mixture, ensuring full coating.

Serve your Classic Baked Brioche Donuts with Cinnamon Sugar warm and enjoy!