Add warm milk, 3 tbsp melted butter, 2 egg yolks and 1/2 tsp vanilla extract, and whisk vigorously to combine well.
Cover with plastic wrap or clean towel and rest at room temperature for 10 minutes, until some bubbles form on the surface.
Add remaining 1 1/2 cups flour, adding just enough until dough holds together and no longer sticks to the sides of the bowl. Knead in the bowl 5 minutes. If dough is really sticking to your hands, dust lightly with flour then continue kneading with clean dry hands. Cover bowl with plastic wrap and let rise in a 100˚F oven for 45 min or at room temperature for 2 hours, or until doubled in size.
Right out of the oven, brush each hot bun all over with the remaining 1 tbsp melted butter and roll in a bowl of sugar to coat lightly. Cut a deep slit in the side of each donut and pipe jam with a pastry bag into each one if desired.