Heat the milk in a cup or bowl in the microwave in 15-second increments or on the stove until it's just warm to the touch. A thermometer should read around 110°F-115°F.

Stir in the yeast and sugar. Loosely cover with plastic wrap and let it stand for 5 minutes until it gets foamy. If the mixture doesn’t get foamy, throw it out and start again with a new packet of yeast.

In the bowl of a stand mixer (or a regular large bowl), whisk together the all-purpose flour, cinnamon, and salt.

Attach the dough hook to the mixer and add the yeast mixture, eggs, butter, and vanilla extract. Mix on medium for 6-8 minutes (or mix by hand) until the dough becomes smooth and soft. The dough will be slightly tacky but not completely sticky. If it is still fairly sticky, knead in 1 tablespoon of additional flour at a time until smooth and elastic.

Coat a large bowl with oil and transfer the dough to the bowl, shaping it into a ball. Cover it with a kitchen towel and let it rise in a warm place until it has doubled in size, about 1 hour.

Line 2 large baking sheets with parchment paper and set aside.

Place the donuts and holes onto the prepared baking sheets, cover them with a clean kitchen towel, and let them rise again for 40 minutes.

While the dough is rising, make the cinnamon sugar topping by mixing together the granulated sugar and cinnamon in a wide and shallow bowl. Set aside.

Line a large plate with paper towels. In a large saute pan or Dutch oven over medium-high heat, heat 2 inches of oil to 360°F. Cook donuts in batches of 2-3 until golden brown on both sides, about 30 seconds per side. The donut holes will cook even faster, so work quickly to prevent any from burning.

Transfer the donuts and holes to the prepared plate to drain and cool for a few seconds, then dip them into the cinnamon sugar and place them on a serving tray. Continue frying and topping the donuts until all are finished.