Dough
1/4 cup warm water
2 tbsp. instant yeast
1 cup warm milk
1/4 cup vegetable shortening
2 eggs
1/4 cup sugar
1 tsp. salt
3 1/2 to 4 cups bread flour
Custard Filling
1/4 cup sugar
3 tbsp. all-purpose flour
3 egg yolks
1 cup plus 2 tbsp. milk
1 tbsp. unsalted butter
1/2 tsp. pure vanilla extract
Vegetable oil for deep-frying
Chocolate Glaze
1 tbsp. unsalted butter
2 ounces semi-sweet chocolate, chopped
1 1/2 cups confectioners' sugar
1/2 tsp. pure vanilla extract
About 1/3 cup hot water
Dough: Hand whisk the water and yeast together in the bowl of an electric mixer; let stand 2 to 3 minutes to dissolve yeast. Whisk in milk and shortening. Stir in eggs, sugar, salt and 3 cups of the flour. Mix to make a soft dough. Knead with the dough hook on lowest speed of mixer for 8 to 10 minutes, gradually adding more flour as required to form a soft but elastic dough. Place dough in a lightly greased bowl, cover the bowl with a damp cloth, and place in a large plastic bag. Close the bag loosely and let dough rise for 30 to 45 minutes, or until almost doubled.
Filling: Whisk sugar, flour and egg yolks together in a bowl. In a small saucepan, bring milk to a boil. Stir 2 tbsp. of the milk into egg mixture to temper the yolks and prevent curdling. Whisk egg mixture into the milk and stir constantly until mixture thickens and begins to boil gently, then immediately remove from heat. Stir in butter and vanilla; let cool. Place in a bowl and cover with plastic wrap, pressing it directly on the surface to keep a skin from forming. Refrigerate.
Turn the dough out onto a lightly floured surface. Pinch off pieces of dough and form into small balls about the size of golf balls. Cover with plastic wrap and let rise for 25 minutes.
In a wok or a Dutch oven, heat 4 inches of oil to 375 degrees F. Add the doughnuts a few at a time and fry for 30 to 40 seconds, until deep brown on the first side. Turn and fry on the second side for 15 to 30 seconds. Using a slotted spoon, transfer to paper towels to drain and let cool.
Whisk the custard to smooth and fluff it. Put custard in a pastry bag fitted with a large plain tip. Split each doughnut in half and pipe or spoon in some custard.
Glaze: Melt the butter and chocolate in a double boiler over barely simmering water. Add confectioners’ sugar and vanilla; stir until smooth. Add enough water to make a thin glaze. Let cool for 1 minute, then drizzle over the doughnuts. If the glaze gets too stiff, add more hot water to make it workable again.