Prepare the Pastry Cream: Heat 2 cups of milk in a saucepan over medium heat until it bubbles. Whisk together 4 egg yolks, ¼ cup of cornstarch, and ½ cup of sugar in a separate bowl. Gradually pour hot milk into the egg mixture while whisking, strain back into the saucepan, and cook until thickened. Stir in 2 tablespoons butter and 1 teaspoon vanilla, then cool completely.

Make Donut Dough: Combine 1 cup warm milk, 2 tablespoons sugar, and 2 teaspoons yeast in a mixing bowl. Let sit for 5 minutes. Add 2 large eggs and ¼ cup melted butter, whisk to combine. Gradually add 3 ½ cups flour and ½ teaspoon salt, mix to form a soft dough, then knead for 5-7 minutes. Place in a greased bowl, cover, and let rise for 1-2 hours.

Shape Donuts: Punch down the dough, roll out to ½ inch thick, cut out donut shapes with a cutter, and place on a lined baking sheet. Re-roll scraps and let them rise for another 30-60 minutes.

Bake Donuts: Preheat oven to 350°F. Bake donuts for 12-15 minutes until golden and springy. Cool on a wire rack.

Fill Donuts: Fill a piping bag with the chilled pastry cream and fill each cooled donut carefully to avoid overstuffing.

Prepare Sugar Topping: Heat ½ cup granulated sugar in a saucepan until melted and golden brown, cool slightly, dip tops of donuts into melted sugar for a crispy topping.