Make the Dough

• In a stand mixer fitted with the dough hook, combine the warm milk, yeast, and sugar. Let it sit for 5 minutes until foamy. If it doesn't foam, your yeast is dead. Add the eggs and vanilla and mix to combine.
• Add the flour and salt and mix until a rough dough forms. Knead for 3–4 minutes on medium-low until the dough begins to smooth out.
• With the mixer running on medium-low, add the butter one tablespoon at a time, waiting until each addition is fully incorporated before adding the next. Once all the butter is in, knead for 8 to 12 minutes until the dough is smooth, elastic, and slightly tacky. Resist adding extra flour during this time — the dough will come together with continued kneading. To test if it's ready, pull off a small piece with greased fingers and gently stretch it. If it stretches thin without tearing, it's ready. If it tears quickly, keep kneading. If the dough is still too sticky, add flour one tablespoon at a time. Avoid adding more than 3 tablespoons.
• Transfer to a greased bowl, cover, and let rise for 1 to 1½ hours until doubled and airy.
• Using greased fingers, turn the dough out onto a lightly floured surface and roll to ½ inch thick. Cut into 3-inch rounds and 1-inch holes. Place on a parchment-lined tray or indivudal parchemnt squares , and let rise for 30 to 45 minutes until puffy.
• Heat 2 to 3 inches of neutral oil in a heavy-bottomed pot to 350°F (175°C). Fry the donuts in batches for 45 to 60 seconds per side until light golden. Transfer to a wire rack.

Make the Glaze:

• Whisk together the powdered sugar, milk, melted butter, vanilla, and salt until smooth and pourable. Set aside.
• While still warm, dip each donut top side down into the glaze, then flip and dip the bottom. Let the excess drip off and place on a rack. Allow the glaze to set for 3 to 5 minutes before serving. These are best enjoyed fresh but can be stored in an airtight container at room temperature for up to 1 day. Reheat briefly in the microwave before serving.