Activate the Yeast: In a small bowl, combine the warm milk, warm water, and a pinch of granulated sugar. Sprinkle the yeast over the top. Stir gently and let it sit for 5-10 minutes until foamy.

Combine Wet Ingredients: In a separate large mixing bowl, whisk together the egg, melted butter, and vanilla extract (if using). Add the activated yeast mixture to these wet ingredients, stirring until well combined.

Add Dry Ingredients: In another bowl, whisk together the flour, remaining granulated sugar, and salt. Gradually add the dry ingredients to the wet ingredients, mixing until a shaggy dough forms.

Knead the Dough: If using a stand mixer, attach the dough hook and knead on low speed for 8-10 minutes, until the dough is smooth, elastic, and pulls away from the sides of the bowl. If kneading by hand, turn the dough out onto a lightly floured surface and knead for 10-15 minutes.

First Rise: Lightly grease a clean bowl with a little oil. Place the dough in the bowl, turning it once to coat. Cover and let it rise in a warm place for 1 to 1.5 hours, or until doubled in size.

Shape the Donuts: Gently punch down the risen dough. Roll out and cut donut shapes using a donut cutter.

Second Rise: Transfer the donuts to parchment-lined baking sheets, cover loosely, and let them rise again for 30-45 minutes.

Prepare for Frying: Heat the oil to 350-375°F (175-190°C).

Fry the Donuts: Carefully lower donuts into the hot oil and fry for 1-2 minutes per side until golden brown.

Drain and Coat: Remove the donuts from the oil and let them drain on paper towels. Coat in sugar and cinnamon if desired.

Serve Immediately: Enjoy fresh and warm donuts.