In a small saucepan, combine the butter, salt, vanilla, and 1 cup water.

Bring to a boil over high heat.

Add the flour and cook, stirring constantly, until a smooth dough forms, about 30 seconds.

Transfer the dough to a bowl and let cool for 1 minute.

Add one egg, stirring vigorously until the dough is smooth.

Repeat with the remaining egg.

Transfer the dough to a floured work surface and pat into a 9-inch square about ¼ inch thick.

Cut the dough square into 16 smaller squares and transfer them to a foil-lined baking sheet.

Dust off excess flour from the dough squares using a dry pastry brush.

Chill the beignets on the sheet in the freezer until frozen solid, at least 1 hour.

Meanwhile, combine the sugar and cinnamon in a large brown paper bag.

Brush 4 squares all over with enough oil to coat well.

Place them in one layer in the air fryer and cook at 400°F until golden brown and puffed, about 13 minutes.

Immediately transfer the cooked beignets to the paper bag and shake them in the cinnamon-sugar to coat.

Repeat with the remaining dough squares and cinnamon-sugar in three more batches.

Serve the beignets hot.