In a large mixing bowl, whisk together the flour, granulated sugar, baking powder, baking soda, salt, and ground nutmeg until well combined.
In another bowl, mix the buttermilk, egg, vanilla extract, and melted butter until fully combined.
Pour the wet ingredients into the dry ingredients and mix until just combined. Do not overmix; the dough should be slightly sticky.
Turn the dough onto a floured surface and knead gently for about 1 minute until it comes together. Pat the dough into a rectangle about 1/2 inch thick.
Cut the dough into squares approximately 2 inches by 2 inches.
In a deep frying pan or pot, heat about 2 inches of oil to 350°F (175°C). Carefully drop the dough squares into the hot oil, being careful not to overcrowd the pan. Fry them for about 2-3 minutes on each side, or until they are golden brown.
Remove the beignets with a slotted spoon and place them on paper towels to drain excess oil.
For the glaze, whisk together the powdered sugar, milk, and vanilla extract until smooth. Adjust the consistency with more milk or powdered sugar as needed.
While the beignets are still warm, dip them into the glaze, making sure they are fully coated.
Allow the glazed beignets to set on a wire rack for a few minutes, then serve warm.
Notes
Serve the beignets on a large plate dusted with powdered sugar and a side of extra glaze for dipping. Optionally, garnish with edible flowers or fresh mint leaves for an added touch of elegance.