In a mixing bowl, combine flour, sugar, yeast, and salt. In a small pot, heat milk and butter until the butter melts. Stir in vanilla extract.
Pour the milk mixture into the dry ingredients and mix until a dough forms. Knead until smooth and elastic, about 5 minutes.
Place the dough in an oiled bowl, cover, and let it rise until doubled in size, about 1-2 hours.
Roll out the dough to 3/4-inch thickness, cut it into donut shapes, and let rise again for 30 minutes. Heat oil to 350°F and fry each donut until golden. Drain on paper towels.
Mix powdered sugar and cocoa powder with 2 tablespoons of milk. Add more milk as needed to achieve a smooth, dipping consistency. Dip each donut into the glaze and set aside to dry.