1. In a medium bowl, whisk together the flour, baking powder, and salt.
2. In a separate bowl, cream together the butter and sugar until light and fluffy. Beat in the eggs one at a time, then mix in the milk and vanilla.
3. Gradually add the dry ingredients to the wet ingredients and mix until just combined. Cover the dough and refrigerate for at least 1 hour.
4. Roll the dough out onto a lightly floured surface to 1/4 inch thick. Cut it into circles with a round cookie cutter.
5. Heat oil to 375 degrees in a deep fryer or large pot. Carefully drop the doughnuts into the hot oil and fry for 1-2 minutes on each side, until golden brown.
6. Remove the doughnuts from the oil and place on a paper towel-lined plate to cool.
7. To make the glaze, whisk together the powdered sugar, milk, and vanilla until smooth. Dip the doughnuts into the glaze and let cool before serving.
