In a large mixing bowl, combine warm whole milk, active dry yeast, and granulated sugar. Whisk in eggs and melted unsalted butter until smooth. Gradually add flour and salt, mixing until a soft dough forms. Knead on a floured surface for about 5-7 minutes until smooth and elastic.

Place the kneaded dough in a greased bowl, covering it with a clean kitchen towel. Allow it to rise in a warm area for about 1 hour or until it has doubled in size.

Punch down the dough gently to release air. Roll it out on a floured surface to about ½-inch thickness. Cut out doughnuts and hole centers. Place them on a parchment-lined baking sheet and let them rest for about 30 minutes.

Heat vegetable oil to 350°F (175°C). Fry the doughnuts in batches for about 1-2 minutes on each side until golden brown. Transfer to a cooling rack to drain excess oil.

Prepare the glaze by whisking together powdered sugar, vanilla extract, and milk until smooth. Dip the warm doughnuts into the glaze and let the excess drip off before placing them back on the cooling rack.