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

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

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

4. Turn the dough onto a lightly floured surface and gently roll it out to about 1/2 inch thick.

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

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

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

8. Transfer the fried beignet squares to a wire rack or paper towel-lined tray to drain slightly.

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

10. Dip or spoon the glaze over the warm beignet squares and let the excess drip off before serving.