Warm the milk to 110°F and stir in the yeast with a pinch of the sugar; let sit 5 minutes until foamy.

In a stand mixer with the dough hook, combine flour, remaining sugar and salt, then add the yeast mixture, eggs, butter and vanilla.

Knead on medium speed for 7 to 8 minutes until the dough is smooth, elastic and pulls cleanly away from the bowl.

Place the dough in a greased bowl, cover, and let rise in a warm spot for 1 hour, until doubled: a finger dent should spring back slowly.

Punch down and roll out on a floured surface to 1/2 inch thick.

Cut into 2.5-inch rounds, re-rolling scraps once, for about 18 rounds.

Set the rounds on a floured tray, cover loosely, and let rise again 30 to 40 minutes until visibly puffy.

Heat 3 inches of oil in a heavy pot to 350°F, where a cube of bread turns golden in 45 seconds.

Fry in batches of 3 to 4, 2 minutes per side, until deep golden and cooked through, then drain on a wire rack.

Once cool enough to handle, pipe about 1 tablespoon of jam into the side of each doughnut until it feels heavy, then dust with powdered sugar.