1. Make compote: Mix berries and sugar in a small bowl. Let stand, stirring once or twice until sugar has dissolved. Cover and refrigerate until ready to use.
2. Make tapioca: Combine milk, sugar, tapioca and egg in a small saucepan and let stand for 5 minutes. Place over medium heat and cook, whisking constantly, until tapioca comes to a boil. Reduce heat to low and cook, whisking, until thickened, about 1 minute.
3. Scrape into a medium bowl, whisk in butter and vanilla; let stand 20 minutes, stirring occasionally. Serve warm with berries, or press plastic wrap onto surface of pudding, refrigerate and serve cold.