Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Add the flour, milk, yeast, and vanilla and mix for 20 to 30 seconds until no dry spots remain.
Switch to the dough hook attachment and mix and knead the dough on medium speed. With the mixer running, add about 1 tablespoon of the butter and knead until fully incorporated into the dough, 30 to 60 seconds. Scrape the bowl if necessary, then continue adding the butter 1 tablespoon at a time, mixing to incorporate fully before adding the next piece. The dough will be wet and sticky; this is OK.
Once all the butter has been added, knead the dough for another 8 to 10 minutes, until it's smooth and begins to pull away from the sides of the bowl slightly. (The dough won’t clean the sides of the bowl, but it should form a ball around the dough hook.)
Transfer the dough to a lightly greased bowl, cover, and allow it to rise in a warm, draft-free area until doubled in size, about 1 hour.
When the dough has doubled in size, generously dust your work surface with flour. Sprinkle the top of the dough with flour, then transfer it to the surface and gently deflate. Lightly dust the top of the dough and your hands with flour, then pat the dough to a 10" square, about 1/2" thick (a little thinner is OK).
Use a floured 2 3/4" cutter to cut rounds of the dough and transfer them to a parchment-lined baking sheet. Bring the scraps together, pat the dough to about 1/2" thick, and continue to cut rounds until you have a total of 12. (Don’t re-roll the scraps more than two times to ensure the doughnuts maintain a tender texture.)
Cover the baking sheet with a reusable cover or greased plastic wrap and let the pączki rise for 30 to 45 minutes, until puffy, while you prepare the oil.
To fry the pączki: Fill a large, heavy-bottomed pot (like a Dutch oven), with oil so that it’s about 2" deep. Heat it over medium heat until it reaches between 350°F and 360°F on a digital or candy thermometer. Line a baking sheet with paper towels.
Once the pączki are puffy and the oil is at temperature, carefully place three pączki in the oil and fry until deep brown, about 2 to 3 minutes. Use a pair of tongs or a slotted spoon to gently flip the pączki and fry the other side until deeply golden brown or mahogany, another 2 to 3 minutes.
Remove the fried pączki from the oil and place on the paper towel-lined baking sheet. Repeat with the rest of the dough rounds, being sure to bring the oil back up to temperature between batches, until all the pączki are fried.
To finish the pączki: Once fully cooled, toss the pączki in a bowl with the non-melting sugar, coating all sides, then place on a clean, parchment-lined baking sheet.
Use a skewer, sturdy straw, or chopstick to poke a hole in the side of each pączki and insert it about halfway; rotate the tool around a couple of times to make a small cavity in the center.
Fill a pastry bag fitted with a round tip or a squeeze bottle with the jam. Insert the pastry tip or the squeeze bottle into the hole and pipe in a generous 1 tablespoon of jam into each pączki. Serve immediately.
Storage information: Pączki are best enjoyed the same day they are fried. Store any leftovers, at room temperature, for 1 day; freezing is not recommended.
