In a large bowl, dissolve yeast in warm milk. Let sit for 5 minutes until foamy.
Add sugar, melted butter, egg, flour, and salt. Mix until a soft dough forms.
Knead dough on a floured surface for 5-7 minutes until smooth and elastic.
Place dough in a greased bowl, cover, and let rise for 1 hour or until doubled in size.
Roll out dough to about ½ inch thickness. Use a donut cutter to cut out shapes.
Place cut doughnuts on a parchment-lined tray, cover, and let rise for another 30 minutes.
Heat oil in a large pot to 350°F. Fry donuts 1-2 minutes per side until golden brown. Drain on paper towels.
For the glaze, whisk together powdered sugar, milk, and vanilla until smooth.
Dip warm donuts in glaze, let excess drip off, and place on a wire rack to set