Add ingredients to bread machine in exact order listed (milk, water, butter, egg, sugar, salt, flour, yeast).

Select "Dough Cycle" (do not stir).

During the rest cycle, open lid and scrape sides with a rubber spatula if needed.

After cycle ends, transfer dough to a floured surface.

Roll out to 1/2-inch thickness and cut into circles (use a donut cutter or two round cutters).

Place on parchment-lined tray, cover, and let rise 30-40 mins until puffy.

Heat 2-3 inches of oil in a heavy pot to 330°F (165°C).

Fry 3-4 donuts at a time (do not overcrowd) for 1-2 mins per side until golden.

Drain on a wire rack and toss in sugar immediately.

In a saucepan, whisk sugar, cornstarch, and 2 cups milk until smooth.

Heat on medium-low until warm (do not boil).

Temper eggs: Slowly whisk in remaining milk + eggs to prevent scrambling.

Cook, whisking constantly, until thickened (like pudding).

Remove from heat, stir in vanilla and salt.

Cover with plastic wrap (touching the custard to prevent skin) and chill.

Use a skewer or chopstick to poke a hole in the side of each donut and swirl to create space.

Fill a pastry bag with custard (fitted with a round tip) and pipe into donuts until "heavy."