Whisk warm milk, yeast, and 1 tablespoon sugar in a stand mixer bowl. Let bloom 5 minutes until foamy. Add flour, remaining sugar, salt, and eggs. Mix on low with dough hook until shaggy, about 2 minutes. Increase to medium and knead 5 minutes until smooth and elastic.

With mixer running, add softened butter one piece at a time, waiting until each is fully absorbed before adding the next. Knead 5-7 minutes until dough is glossy, soft, and pulls away from bowl cleanly. It will be sticky - resist adding flour.

Transfer to a buttered bowl, cover with plastic wrap, and let rise in a warm spot until doubled, 1 to 1.5 hours. The dough should hold an indent when gently pressed.

Punch down dough and turn onto a lightly floured surface. Roll to 1/2-inch thick. Cut 8 rounds with a 3-inch cutter, rerolling once if needed. Place on parchment-lined baking sheets, cover loosely, and rise 45-60 minutes until puffy and nearly doubled.

Heat oil to 350°F in a heavy pot. Carefully lower 2-3 donuts at a time, parchment and all - it will release easily. Fry 2-3 minutes per side until deep golden brown and cooked through. Drain on a wire rack set over a baking sheet. Toss warm donuts in sugar to coat completely.

Beat cold cream, powdered sugar, and vanilla to stiff peaks. Transfer to a piping bag fitted with a long narrow tip. Poke into each donut from the side and pipe in cream until you feel resistance. Repeat with raspberry jam using a clean bag, aiming for a distinct layer. Serve within 2 hours for best texture.