In a large bowl, mix together all the ingredients until well combined. Let the dough rest for 5 minutes. Knead the dough with an electric mixer fitted with a dough hook or by hand until smooth and soft, about 6 to 8 minutes. Place the dough in a large oiled bowl, cover with plastic wrap and let rise in a warm place until doubled in size, about 1 1/2 to 2 hours.

Punch the dough down and turn it out onto a lightly floured work surface. Roll the dough out into 1/4-inch thickness and cut with a doughnut cutter or with a large round cutter and a small cutter for the holes. Cover the dough with a clean kitchen towel and let rise until doubled in size, about 1 hour.

Heat the oil in a large, deep, and heavy pan to 350°F. Place three doughnuts in the oil at a time and fry until golden brown, about 1 minute per side. Do not overcook the doughnuts. Drain on a paper towel-lined plate.

In a medium saucepan over medium heat, combine the butter, milk, corn syrup, and vanilla and heat until the butter melts. Decrease the heat to low and add the chocolate, whisking until melted. Turn off the heat and add the powdered sugar, whisking until smooth. Immediately dip the doughnuts into the glaze. If the glaze begins to set return the saucepan to low heat and stir until liquid again. Let the glaze set on the doughnuts for 30 minutes before serving.