In a small bowl, stir together the warm water, 1 tsp of caster sugar, and yeast. Let stand for 5-10 minutes until a frothy layer forms on top.

In a stand mixer bowl, combine warm milk, remaining caster sugar, egg, melted butter, and vanilla. Add the yeast mixture, then the flour and salt. Knead with a dough hook for 5-7 minutes until smooth and supple.

Place the dough in a lightly oiled bowl, cover, and let rise in a warm place for 1-2 hours, or until it has doubled in size.

Punch down the dough and roll it out on a floured surface to 1/2-inch thickness. Cut into 2-inch squares. Place on a floured tray, cover, and let rest for 15-20 minutes.

Heat 2-3 inches of oil in a heavy-based pan to 180°C (350°F). Fry the beignets in small batches for 1-2 minutes per side until golden brown and puffed up.

Remove beignets to a wire rack lined with paper towels. While still hot, dust them generously with icing sugar and serve immediately.