Mix warm milk and yeast in stand mixer bowl. Add eggs, sugar, salt, and flour. Use dough hook on low until dough forms.
Gradually add room temp butter while mixing. Increase to medium-high, mix for 3 mins until smooth. Cover, let dough double.
Combine brown sugar, sugar, cinnamon in bowl. Set aside.
Transfer dough to floured surface, shape into round. Cut into 12 pieces, form each into ball. Cover, let rise until doubled.
Heat oil in pot to 360°F. Fry doughnuts few at a time, 2 mins per side until golden. Drain on paper towels, coat with vanilla sugar.
Cool slightly, use knife to create filling pockets.
Fill with jam, cream, or curd using pastry bag with large tip.
Optionally dip in ganache or glaze. Customize toppings to taste.