Beat ¾ cup milk with the yeast, flour, butter, white sugar, salt, and egg on medium speed for 10 minutes. You can also knead it by hand until smooth.
Turn the dough out onto a lightly-floured surface, and shape it into a ball. Put the dough in a bowl, and cover it with a kitchen towel. Let the dough rest at room temperature for 1 hour.
Return the dough to the floured surface, and roll it out until it's ½-inch thick. Cut the dough into 6 5-inch circles, and then cut 1-inch holes in the center.
Heat the oil over medium heat to 375 F. Fry each doughnut for around 1 minute on each side, or until golden brown, working in batches if necessary to avoid overcrowding. Cool the donuts on a wire rack.
Mix the remaining ¼ cup of milk with the powdered sugar, corn syrup, and vanilla, stirring until smooth. Dip each doughnut into the glaze, then return to the cooling rack until the glaze hardens.