Activate the yeast (if using active dry): In a small bowl, mix warm milk with 1 teaspoon of the sugar. Sprinkle in the yeast, stir, and let it sit until foamy, 5–10 minutes. If using instant yeast, skip this step and add it directly to the flour.

Make the dough: In a large bowl, combine flour, remaining sugar, and salt. Add eggs, vanilla, and milk (with yeast if proofed). Mix with a wooden spoon or stand mixer until a shaggy dough forms.

Knead and enrich: Knead by hand on a lightly floured surface or with a dough hook for 6–8 minutes, until smooth. Add softened butter, a little at a time, kneading until the dough is elastic and slightly tacky but not sticky. This may take another 4–6 minutes.

First rise: Shape into a ball, place in a lightly oiled bowl, and cover. Let rise in a warm spot until doubled, about 1–1.5 hours. For a slow rise, refrigerate overnight and continue in the morning.

Shape the donuts: Turn the dough out onto a lightly floured surface. Gently roll to about 1/2 inch (1.25 cm) thick. Cut rounds using a 2.5–3 inch (6–7.5 cm) cutter. Gather scraps, rest them for 10 minutes, and cut more.

Second rise: Place rounds on parchment-lined trays, leaving space between them. Lightly cover and let rise until puffy, 30–45 minutes. They should look airy and soft when gently poked.

Heat the oil: In a heavy pot, heat 2–3 inches (5–7.5 cm) of oil to 350–355°F (175–180°C). Use a thermometer—this matters for light donuts.

Fry: Fry 3–4 donuts at a time, about 1–2 minutes per side, until golden. Adjust heat to keep oil in range. Transfer to a paper towel–lined rack to drain.

Fill: Fit a piping bag with a small round tip. Poke a hole in the side of each donut with a skewer, then pipe in jam until you feel the donut slightly swell. Aim for about 1–2 teaspoons per donut depending on size.

Finish: Dust generously with powdered sugar while still slightly warm. Serve fresh for the best texture.