Whisk the gluten-free bread flour, instant yeast, sugar, and salt in a large bowl.
Add the warm milk, egg, melted butter, and vanilla. Beat with a sturdy spoon or mixer for about 2 minutes. The dough should be soft and slightly sticky, not dry.
Cover the bowl and let the dough rise in a warm place for 45–60 minutes. Gluten-free dough may expand without fully doubling.
Place the dough between two lightly floured sheets of parchment. Press or roll to about 1/2-inch thick, then cut into twelve 2-inch squares.
Preheat the air fryer to 350°F for about 3 minutes if your model benefits from preheating.
Lightly spray the basket and arrange the dough squares in one layer with space between them. Mist the tops lightly. Air fry at 350°F for 6–8 minutes, turning halfway, until puffed and golden. Cook in batches if needed.
Brush the warm beignets lightly with melted butter if desired and dust generously with powdered sugar. Serve immediately.