Combine milk, boiling water, and yeast in a large bowl. Let stand for 5 minutes until foamy.

Add flour, sugar, egg, butter, and salt to the yeast mixture. Knead the dough until smooth and elastic (about 8-10 minutes).

Place the dough in a lightly oiled bowl, turning to coat. Cover and let rise in a warm place for 1-1.5 hours, or until doubled in size.

Punch down the dough and roll it out to about 1/2 inch thickness. Use a donut cutter or a biscuit cutter and a small knife to cut out donuts and donut holes.

Heat oil in a wok or deep fryer to 350°F (175°C). Carefully fry the donuts for 1-2 minutes per side, or until golden brown.

Remove donuts and place them on a wire rack to drain excess oil. Let cool slightly.

While donuts are cooling, prepare the glaze: Whisk together butter, confectioners' sugar, vanilla, and hot water until smooth. Add more hot water if needed to reach desired consistency.

Dip the cooled donuts in the glaze, ensuring they are fully coated. Place them back on the wire rack to let the glaze set.