Prepare the donuts: In a large bowl, whisk together flour, sugar, baking powder, and salt.
In a medium bowl, whisk together milk, eggs, melted butter, and vanilla extract.
Gradually add wet ingredients to dry ingredients, stirring until just combined. Be careful not to overmix.
Cover the dough and let it rest for 10 minutes.
On a lightly floured surface, roll out the dough to about 1/2 inch thickness. Use a donut cutter to cut out the donuts and donut holes.
Heat oil in a deep fryer or large pot to 375°F (190°C).
Fry the donuts for about 1-2 minutes per side, or until golden brown.
Remove donuts from the oil and place them on a wire rack lined with parchment paper to drain excess oil.
Make the chocolate glaze: In a small bowl, whisk together powdered sugar, cocoa powder, milk, and vanilla extract until smooth. Adjust consistency by adding more milk if needed.
Once donuts are completely cooled, dip the tops in the chocolate glaze.
Let the glaze set before serving.