1. In a large bowl, whisk together the flour, granulated sugar, baking powder, baking soda, and salt.

2. In a separate bowl, whisk the buttermilk, egg, melted butter, and vanilla extract until smooth.

3. Pour the wet ingredients into the dry ingredients and stir just until a soft dough forms.

4. Turn the dough onto a lightly floured surface and roll it into a rectangle about 1/2 inch thick.

5. Cut the dough into even squares using a sharp knife or bench scraper.

6. Heat 2 inches of vegetable oil in a heavy pot to 350°F.

7. Fry the squares in batches for 1 to 2 minutes per side, until puffed and golden brown.

8. Transfer the fried squares to a paper towel-lined plate or wire rack.

9. In a medium bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth.

10. Drizzle or dip the warm beignet squares in the glaze.

11. Let the glaze set slightly, then serve warm or at room temperature.