Warm the milk until lukewarm. Whisk in a spoonful of the sugar, then sprinkle in yeast. Rest until foamy.
Beat eggs, remaining sugar, vanilla, salt, and softened butter until combined.
Stir in the yeast mixture. Add flour gradually until a soft dough forms.
Knead on a lightly floured counter for 8 minutes, until smooth and stretchy.
Place in a greased bowl, cover, and let rise until doubled, about 60–75 minutes.
Punch down the dough. Roll to 1/2 inch thickness and cut 3-inch rounds. Re-roll scraps.
Set rounds on parchment, cover, and proof until puffy, 30–45 minutes.
Heat oil to 350°F in a deep pot. Fry doughnuts 2–3 minutes per side until deep golden.
Drain on a rack. While warm, toss each doughnut in granulated sugar to coat.
Mix strawberry preserves with lemon juice. Spoon into a pastry bag fitted with a round tip.
Pierce the side of each doughnut with the tip and pipe in jam until slightly heavy.
Serve warm or at room temperature. Best enjoyed the day they’re made.