Make the dough: In a mixing bowl of a stand mixer, whisk the warm milk, yeast, and sugar together. Let it sit for 10 minutes until foamy on the surface.
Add the eggs, melted butter, vanilla, salt, and flour. Mix on low-medium speed for about 3 minutes until a soft ball of dough starts to form. Scrape down the sides of the bowl with a rubber spatula as needed. If it’s too sticky, add 2 tablespoons at a time, and knead until the dough comes together and pulls away from the sides of the bowl. The dough will be just slightly sticky and workable. Keep kneading for 7-8 minutes or knead by hand on a floured surface for 8-10 minutes. Place dough in a lightly greased bowl and cover with plastic wrap. Let rest in a warm place for about 1.5 hours, or until doubled in size.
After the dough reaches doubled size, punch it down to remove the air. On a lightly floured surface, roll dough into a rectangle with 1/2-inch thickness. Cut into doughnuts using a doughnut cutter or with a 2¾-inch (7cm) round cutter and cut holes from the middle of the donuts and with a ¾-inch (2cm) piping tip. You can use any cutter that you like depending on your preferred donut size. Press together scraps, re-roll, and cut out more doughnuts. Place doughnuts on a baking sheet lined with parchment paper, leaving space between. Cover loosely with plastic wrap and let rest to rise for about 45-60 minutes, or until doubled in size.
Make the cinnamon sugar mixture: In a small bowl, toss sugar and cinnamon together. Set aside.
Frying donuts: In a large, heavy-bottomed saucepan, heat oil until it reaches 350F/175C. If you don’t have a thermometer to test, stick the end of a wooden spoon into the oil. It’s ready to fry when many bubbles form around the stick. Carefully, gently add doughnuts to hot oil, two to three at a time, depending on how large your saucepan is. Deep fry until golden brown, about 1-1.5 minutes on each side. Remove from the oil, using a slotted spoon and place on a cooling rack. You can drain donuts using a paper towel if desired. While donuts are still warm, toss them in cinnamon-sugar mixture. Repeat the process with the rest of the donuts.
Donuts are best eaten the same day they are made. Store leftovers in an airtight container at room temperature.