Separate eggs and beat egg whites until stiff. Gradually beat half the sugar into egg whites until shiny and stiff peaks form. Beat butter until creamy and gradually beat in remaining sugar. Halve vanilla bean lengthwise, scrape out seeds with a sharp knife and beat seeds and egg yolks into butter mixture. Mix flour, baking powder and salt. Carefully fold egg whites into egg yolk mixture. Sift flour mixture over egg mixture and fold in alternately with milk. If necessary, add more milk or flour to form pliable dough.
Heat oil in a deep fryer or pan until bubbles rise to surface, about 160°C (approximately 325°F). On a floured surface, divide dough into 20-25 equal pieces, shape into rolls and connect ends to form rings. Fry doughnuts in batches until golden brown on both sides, 5-10 minutes. Roll doughnuts in sugar while still hot. If desired, sprinkle doughnuts with cinnamon to serve.