In a small bowl, combine milk and 1 tsp. sugar, stir to dissolve. Sprinkle yeast over mixture and let sit until frothy, about 8 minutes.
In a large bowl, whisk together remaining ¼ cup sugar, butter, egg and vanilla. Pour in yeast mixture, add flour and salt, and stir to combine.
Transfer dough to a lightly floured surface and knead until elastic and only slightly tacky. Form into a ball. Grease a large bowl with cooking spray, add dough, and cover with a clean dish towel. Let dough rise in a warm spot until doubled in size, about 1 hour.
Line a large baking sheet with parchment paper and grease lightly with cooking spray. Punch down dough, then turn onto a lightly floured work surface and roll out into a 5cm-thick rectangle.
Using a donut cutter or 7cm and 2.5cm biscuit cutters, punch out your doughnuts. Knead the scraps together, reroll and punch out more doughnuts. Place doughnuts and holes onto baking sheet, cover with a dish towel and let rise again, 40 minutes to 1 hour more.
Grease air-fryer basket with cooking spray, and add 2 donuts and 2 donut holes at a time, making sure donuts don’t touch. Cook at 190° Celsius until deeply golden, 6 minutes. Move to cooling rack and repeat with remaining dough.
Dip doughnuts in glaze (see below) or follow instructions for cinnamon sugar. Return to cooling rack and let set for 5 minutes before serving.