In a large mixing bowl, whisk the yogurt, eggs, sugar, and vanilla until smooth. Add the self-rising flour and mix until the dough is thick and free of lumps.
Heat oil in a deep pot to 350°F/180°C. Using an ice-cream scoop or a spoon, drop small portions of dough into the hot oil. Fry until golden on all sides, then transfer to a paper-towel-lined tray.
Fill a piping bag with your chosen spread and pipe into each donut. Dust with powdered sugar and serve warm.