Make the dough: In a small bowl, combine warm milk, yeast, and sugar. Let stand for 5 minutes, or until foamy.
In a large bowl, whisk together the egg, melted butter, and vanilla extract.
Add the yeast mixture to the egg mixture and stir to combine.
In a separate bowl, whisk together the flour and salt. Gradually add the dry ingredients to the wet ingredients, mixing until a dough forms.
Turn the dough out onto a lightly floured surface and knead for 5-7 minutes, or until smooth and elastic.
Place the dough in a greased bowl, cover with plastic wrap, and let rise in a warm place for 1 hour, or until doubled in size.
Punch down the dough and divide it into 12 equal pieces. Roll each piece into a long, thin rope.
Twist the rope into a cruller shape and place it on a ungreased baking sheet. Repeat with the remaining dough.
Let the donuts rise for 30 minutes, or until doubled in size.
Preheat oven to 375°F (190°C).
Bake the donuts for 15-20 minutes, or until golden brown.
Let the donuts cool on a wire rack for a few minutes.
Make the glaze: In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth.
Dip the cooled donuts into the glaze and let them set before serving.
