1. Activate the yeast by combining lukewarm whole milk with a bit of granulated sugar in a small bowl. Sprinkle the yeast over the mixture and let it sit for about 10 minutes until bubbly and frothy.
2. Combine all-purpose flour, remaining sugar, and salt in a large mixing bowl. Add the yeast mixture, eggs, and vanilla extract to the dry ingredients. Knead by hand or with a dough hook attachment until well mixed.
3. Gradually add softened butter while kneading the dough, ensuring it fully incorporates. Continue kneading until the dough is smooth and elastic, passing the windowpane test.
4. Transfer the kneaded dough to a lightly greased bowl. Cover with a clean kitchen towel and proof in a warm environment until doubled in size, about 1-2 hours, or refrigerate overnight for a slower rise.
5. Once proofed, punch down the dough gently to release air. Roll it out on a floured surface to half an inch thick and use a round cutter to shape donuts.
6. Place the cut donuts on a parchment-lined baking sheet. Cover with a towel and let them proof again until slightly puffed, 30-60 minutes.
7. Heat neutral oil, such as canola or vegetable oil, in a large pot or Dutch oven to 350°F (180°C). Prepare a cinnamon sugar mixture in a shallow bowl for coating.
8. Carefully fry the donuts, a few at a time, for about 90 seconds per side until golden brown. Use a slotted spoon to transfer them to a wire rack to drain excess oil.
9. While still warm, toss the donuts in the prepared cinnamon sugar mixture to coat evenly.
10. Optional: If desired, fill the cooled donuts with Bavarian cream, jam, or chocolate ganache using a pastry bag before serving.
11. Serve warm or store unfilled donuts in an airtight container for up to 3-4 days at room temperature or a week in the fridge. Filled donuts should be refrigerated.
12. Enjoy your homemade Cinnamon Sugar Brioche Donuts and share with family and friends!