Activate Yeast: In a small bowl, combine lukewarm milk, granulated sugar, and active dry yeast. Stir gently and let sit for 5 to 10 minutes until frothy.

Mix Wet Ingredients: Add melted unsalted butter, large egg, and vanilla extract to the yeast mixture. Blend thoroughly until smooth.

Combine Dry Ingredients: In a separate bowl, whisk together the flour and salt, then gradually add to wet ingredients, mixing until a shaggy dough forms.

Knead Dough: Turn dough onto a floured surface and knead for 5 to 10 minutes until smooth and elastic. Let rise in a greased bowl covered for 1 to 1.5 hours.

Shape Donuts: After rising, punch down the dough, roll it out to ½-inch thickness and cut out circles. Let them rise again for 30 to 45 minutes.

Heat Oil: Heat vegetable oil in a deep pot to 350°F, monitoring with a thermometer.

Fry Donuts: Drop donut holes into the hot oil in batches, frying 1 to 2 minutes per side until golden brown. Transfer to paper towels.

Prepare Glaze: Melt semi-sweet chocolate and unsalted butter over a double boiler, stir until smooth. Add milk, vanilla extract, salt, and sifted powdered sugar.

Glaze Donuts: Dunk warm donut holes into the glaze, allowing excess to drip off and place on a wire rack to set.