Prepare Dough: In the bowl of a stand mixer, add milk, yeast, 50g sugar, salt, eggs, and flour. Mix on low for 7-8 minutes until the dough comes together.
Add Butter: Continue mixing on low while adding the butter, 2 tablespoons at a time, until fully absorbed.
First Rise: Transfer dough to a well-floured surface, form into a ball, then place in a greased bowl. Cover and let rise until doubled in size, about 2-4 hours.
Prepare for Second Rise: Cut greaseproof paper into squares and set aside. Roll out dough to about 2.5cm thickness and cut out rounds. Place on greaseproof paper and let prove for another hour.
Fry Doughnuts: Preheat oil to 160°C (320°F). Mix 100g sugar with cinnamon. Fry doughnuts until golden brown, 2-3 minutes per side.
Coat and Fill: Drain doughnuts and roll in cinnamon sugar mixture. Let cool, then fill with jam using a piping bag.