Activate the yeast: In a large bowl, combine warm milk, granulated sugar, and yeast. Let stand for 5-10 minutes until foamy.
Combine wet ingredients: In a separate bowl, whisk together melted butter and eggs.
Mix dry ingredients: In a large bowl, whisk together flour and salt.
Bring together: Gradually add the wet ingredients to the dry ingredients, mixing until a dough forms.
Knead: Turn the dough out onto a lightly floured surface and knead for 5-7 minutes until smooth and elastic.
First rise: Place the dough in a greased bowl, cover with plastic wrap, and let rise in a warm place until doubled in size (about 1 hour).
Shape donuts: Punch down the dough and roll it out to about 1/2 inch thickness on a lightly floured surface. Cut out donut shapes using a donut cutter.
Second rise: Place the donut shapes on a parchment-lined baking sheet, cover with plastic wrap, and let rise for 30 minutes.
Fry donuts: Heat vegetable oil in a large pot to 375°F (190°C). Carefully fry the donuts for 1-2 minutes per side, or until golden brown. Remove and drain on paper towels.
Make the glaze: In a small bowl, whisk together powdered sugar, milk, and vanilla extract until smooth.
Glaze donuts: Dip the cooled donuts in the glaze and let excess drip off. Place on a wire rack to set.