Scald the milk: In a small saucepan over medium-low heat, warm the milk, stirring constantly, until it reaches 180°F or begins to steam. Remove from heat and let cool to 115°F on a thermometer.

Proof the yeast: In a large mixing bowl, whisk together 1 cup of flour, sugar, yeast, and salt. Pour in the warm milk, then whisk in the olive oil, egg yolks, and vanilla extract until smooth. Cover the bowl with plastic wrap and let rest at room temperature for 10 to 15 minutes, until bubbles appear on the surface.

Knead the dough: Using a firm spatula, stir in the remaining 1 1/2 cups of flour, adding just enough until the dough comes together and no longer sticks to the bowl. If needed, add the last 2 tablespoons of flour. Knead the dough by hand in the bowl for about 5 minutes, dusting your hands lightly with flour if it's too sticky. The dough should be soft and slightly tacky.

First rise: Cover the bowl with plastic wrap and set in a warm place, such as a 100°F oven, for 45 minutes, or at room temperature for 1 1/2 to 2 hours, until the dough has doubled in size.

Roll and cut: Turn the dough out onto a lightly floured surface. Roll it to about 1/4 to 1/3 inch thick. Use a donut cutter to cut out 11 to 12 donuts and donut holes. Gently press the scraps together and re-roll to cut additional donuts if desired.

Second rise: Place the donuts and donut holes on a parchment-lined baking sheet, spacing them so they aren't touching. Cover with a clean tea towel and let rise in a warm spot for 20 minutes, or at room temperature for 45 minutes, until puffed.

Make the glaze: In a medium bowl, whisk together the powdered sugar, 5 tablespoons of water, and vanilla extract until smooth. Set aside.

Heat the oil: In a Dutch oven or deep fryer fitted with a clip-on thermometer, heat about 2 inches of oil to 375°F.

Fry the donuts: Once the donuts have puffed, carefully fry 2 at a time for 45 to 60 seconds per side, flipping once, until golden brown. Use a slotted spoon to transfer them to a paper towel-lined plate for just a few seconds.

Glaze: While still warm, dip each donut into the glaze, turning to coat both sides. Place on a wire rack set over a baking sheet to let the excess drip off.

Fry the donut holes: Fry the donut holes in 2 batches for about 30 to 45 seconds total, then glaze them as well. Serve the donuts once the glaze has set.