Step 1

Microwave the milk in a small bowl until 105 to 110 degrees F. Stir in the yeast and 1 tablespoon of sugar; let stand until foamy, about 5 minutes. Whisk together the flour and salt in the bowl of a stand mixer fitted with the dough hook.

Step 2

Add the yeast mixture, butter, vanilla, zest, egg, egg yolks and remaining 1/2 cup sugar to the flour mixture. Mix with a spoon until the dough just comes together, about 1 minute. (It's OK if a little of the flour remains loose at the bottom of the bowl, it will all come together while kneading.) Knead the dough on medium speed until the dough is smooth and elastic, about 8 minutes. Turn the dough out onto a lightly floured surface and knead into a smooth ball, about 1 minute. Lightly spray the mixing bowl with nonstick cooking spray and return the dough to the bowl; cover the bowl tightly with plastic wrap and let rise in a warm place until it has doubled in size, 1 1/2 to 2 hours.

Step 3

Turn the dough out onto a floured surface and gently roll out into a 1/2-inch-thick round. Using a 3-inch round cookie cutter or drinking glass, cut the rounds out as closely together as possible. Place the rounds about 2 inches apart on a baking sheet lightly sprayed with nonstick cooking spray.

Step 4

Lightly spray the tops with the cooking spray. Cover with plastic wrap and let rise again until puffed, at least 30 minutes and up to 1 hour. The dough rounds may not increase in size during this proofing, which is fine. The point is to let the dough rest before frying, resulting in pillowy doughnuts.

Step 5

To fry the doughnuts: Line a baking sheet with a double layer of paper towels for draining. Put 1/2 cup granulated sugar in a medium bowl for coating the fried doughnuts.

Step 6

Fill a large, heavy bottomed pot with 3 inches vegetable oil (about 8 cups) and insert a deep-fry thermometer. Place over medium-high heat and bring the oil temperature to 345 to 350 degrees F. Do not let the oil get any hotter than 350 degrees or the outside of the doughnuts will burn before the insides are cooked through.

Step 7

Working in batches of 3 to 4, carefully slip the dough rounds into the hot oil, being careful not to let the oil splash up. Fry the doughnuts until deep golden brown, about 2 minutes. Use a slotted spoon to gently flip the doughnuts over and fry until deep golden brown on the other side, about 2 minutes. Use a slotted spoon to transfer the doughnuts the paper towel-lined baking sheet. Return the oil to temperature between batches.

Step 8

Roll the warm doughnuts in the sugar to coat, then transfer them back to the baking sheet. Using a chopstick or skewer, poke a hole in the top of the donut, wiggling the stick back and forth to form a little pocket. Fill a pastry bag fitted with a small round tip with the jam of your choice and gently squeeze about 2 tablespoons into the side of each doughnut until the jam just peaks out from the hole on top. Dust the filled doughnuts with confectioners' sugar, if using, and serve hot.