Bloom the yeast (if using active dry): Stir the yeast and a teaspoon of sugar into the warm milk. Let it sit 5–10 minutes until foamy. If using instant yeast, skip the bloom and add it directly to the flour.
Make the dough: In a large bowl, whisk flour, sugar, and salt. Add the milk-yeast mixture (or milk if using instant yeast), eggs, and vanilla. Mix until a shaggy dough forms. Knead by hand or with a mixer on medium-low for 2–3 minutes.
Add the butter: Knead in the softened butter a tablespoon at a time until the dough is smooth, elastic, and slightly tacky, about 6–8 more minutes. It should pull away from the bowl but still feel soft.
First rise: Shape into a ball and place in a lightly oiled bowl. Cover and let rise in a warm spot until doubled, about 1–1.5 hours, depending on your kitchen temperature.
Shape the doughnuts: Turn the dough onto a lightly floured surface. Gently pat to about 1/2 inch (1.25 cm) thick. Use a 2.5–3 inch (6–7.5 cm) round cutter to cut circles. Gather scraps and repeat. Place rounds on parchment-lined baking sheets, leaving space between them.
Second rise: Cover loosely with a clean towel and let rise until puffy, about 30–45 minutes. They should look light and slightly domed when ready.
Heat the oil: In a heavy pot, heat 2–3 inches (5–7.5 cm) of oil to 350–360°F (175–182°C). Keep a thermometer clipped on to maintain temperature. Prepare a tray with paper towels and a shallow bowl of sugar (with cinnamon if using).
Fry in batches: Carefully lift dough rounds and lower into the oil, a few at a time. Fry about 1–2 minutes per side until deep golden, turning once. Avoid crowding the pot.
Drain and coat: Transfer to paper towels for 10–15 seconds, then roll warm doughnuts in sugar to coat all sides. Work while they’re still warm so the sugar sticks well.
Fill with jam: Fit a piping bag with a long, narrow tip (Bismarck tip) and fill with jam. Pierce the side of each doughnut and gently pipe in about 1–2 tablespoons of jam until it feels plump. No piping bag? Use a small knife to make a slit and spoon or squeeze jam in with a zip-top bag.
Serve: Enjoy slightly warm or at room temperature. They’re best the day they’re made.