1. Make the pastry cream by heating milk until steaming. Whisk egg yolks, sugar, and cornstarch until smooth. Slowly add hot milk to temper and then cook until thickened. Stir in butter and vanilla, chill.
2. For the dough, dissolve yeast in warm milk with a pinch of sugar. Add remaining sugar, egg, vanilla, and butter. Mix in flour and salt to form a dough.
3. Knead the dough until smooth and let rise in a greased bowl until doubled (about 1 hour).
4. Roll out dough to ½ inch thickness, cut donuts with a round cutter. Let rise again until doubled.
5. Bake at 350°F for 10–12 minutes until golden. Cool completely.
6. Fill donuts with chilled pastry cream using a filling tip.
7. Prepare caramel by heating sugar, water, and salt until amber. Dip donut tops into caramel for a crunchy shell.
8. Allow caramel to harden, then serve.