Microwave the milk in a small bowl until 105 degrees F to 110 degrees F. Stir in the yeast and 3 tablespoons sugar; let stand until foamy, 3 to 5 minutes. Whisk the flour and salt in the bowl of a stand mixer fitted with the dough hook.
Add the yeast mixture, eggs and vanilla to the flour. Mix on medium speed until the dough comes together, 1 to 2 minutes. Mix in the butter 1 piece at a time. Continue mixing, scraping the bowl occasionally with a rubber spatula, until the dough is smooth and elastic, about 6 minutes (it will be a little sticky). Lightly coat a large bowl with cooking spray. Scrape the dough into the bowl, cover tightly with plastic wrap and let rise in a warm place until doubled in size, about 1 hour.
Line 2 baking sheets with parchment paper; dust with flour.
Turn out the dough onto a well-floured surface and gently pat until 1/2 inch thick. Using a 3-inch round cutter, cut out 12 to 14 rounds as close together as possible. Place the rounds 1 1/2 inches apart on the prepared pans. Lightly coat the tops of the rounds with cooking spray. Cover with plastic wrap and let rise until puffy, 30 to 45 minutes.
Heat 2 inches of vegetable oil in a heavy pot until a deep-fry thermometer registers 350 degrees F. Cut the parchment paper around each dough round so they can be picked up separately. Working in batches of 3 or 4, use the parchment to carefully slide the dough rounds into the hot oil. Fry the doughnuts, flipping occasionally, until golden brown, 2 1/2 minutes. Using a slotted spoon, transfer the doughnuts to a paper towel?lined baking sheet to drain. Return the oil temperature to 350 degrees F between batches.
Put the remaining 1/2 cup sugar in a medium bowl. While the doughnuts are still warm, toss them in the sugar to coat, then transfer to a rack to cool completely.
Using a chopstick or skewer, poke a hole in the side of each doughnut, wiggling the stick around to form a pocket. Fill a pastry bag fitted with a small round tip with the jelly and squeeze about 2 tablespoons into each doughnut.
Blueberry-Lemon Poppy Seed
Skip the sugar coating in Step 5. Fill the doughnuts with blueberry jam. Top with lemon glaze (whisk 1 cup confectioners? sugar with the zest of 1 lemon, 2 tablespoons lemon juice and a pinch of salt). Sprinkle with poppy seeds.
In Step 5, coat the doughnuts with cinnamon sugar (whisk 3/4 cup sugar with 1 tablespoon ground cinnamon). Fill the doughnuts with apple butter.
Learn how to make a Hanukkah favorite: jelly doughnuts!
In Step 5, coat the doughnuts with raspberry sugar (pulse 1/2 cup freeze-dried raspberries in a food processor or spice grinder until powdery; mix with 1/3 cup sugar). Fill the doughnuts with lemon cream (whisk 1/2 cup heavy cream to stiff peaks, then fold in 1 cup lemon curd).
Skip the sugar coating in Step 5. Fill the doughnuts with strawberry jam. Top with chocolate glaze (melt 6 ounces chopped semisweet chocolate with 1/3 cup heavy cream and a pinch of salt in the microwave, then stir until smooth).