Bloom the yeast: Warm the milk to about 110°F/43°C—comfortable to the touch, not hot.
Stir in the yeast and a pinch of the sugar. Let it sit for 5–10 minutes until foamy. If it doesn’t foam, start over with fresh yeast.
Make the dough: In a large bowl, whisk the remaining sugar, salt, and eggs.
Add the bloomed yeast mixture and vanilla. Stir in 3 1/2 cups of flour to form a shaggy dough. Knead by hand or with a mixer on medium-low for 2–3 minutes.
Add the butter: Knead in the softened butter, 1 tablespoon at a time, until fully absorbed.
The dough will be sticky at first; keep mixing for 6–8 minutes until smooth, elastic, and slightly tacky but not messy. Add up to 1/2 cup more flour as needed.
First rise: Shape the dough into a ball and place it in a lightly greased bowl. Cover and let it rise in a warm spot until doubled, about 1–1 1/2 hours.
Roll and cut: Turn the dough onto a lightly floured surface.
Roll to about 1/2 inch thick. Cut rounds with a 2 1/2 to 3-inch cutter. Gather scraps, rest them 5 minutes, then re-roll and cut again.
Second rise: Place rounds on parchment-lined baking sheets, leaving space between.
Cover lightly with plastic or a clean towel. Let them rise until puffy and slightly jiggly, 30–45 minutes.
Heat the oil: In a heavy pot, heat 2–3 inches of oil to 350°F/175°C. Keep a thermometer in the pot for accuracy.
Prepare a tray with paper towels and a shallow bowl with granulated sugar for rolling.
Fry the donuts: Fry 3–4 at a time to avoid crowding. Cook 1–2 minutes per side until golden brown. Adjust heat to maintain 350°F.
Transfer to paper towels to drain briefly, then roll warm donuts in sugar. Repeat with all donuts.
Prep the filling: Stir the jam until smooth. If it’s very thick, warm it slightly or whisk in a teaspoon of water to loosen.
Load into a piping bag fitted with a Bismarck tip, or use a zip-top bag with a small hole snipped in a corner.
Fill the donuts: Use a skewer to make a tunnel from the side into the center of each donut. Insert the piping tip and squeeze in about 1–2 tablespoons of jam. You’ll feel the donut gently expand.
Wipe any excess and add a final roll in sugar if needed.
Serve: Enjoy warm or at room temperature. The centers should be bright and jammy, with a soft, airy crumb and crisp sugar shell.