In a large bowl, combine 1/4 cup sugar, yeast, lukewarm milk, and egg. Add flour, softened butter, and salt. Mix until a soft dough forms.
Turn the dough onto a floured surface and knead until smooth and elastic, about 5 minutes.
Place the dough in a greased bowl, cover with a cloth, and let rise in a warm place until doubled in size, about 1 hour.
Roll out the dough to 1/2 inch thickness. Cut out donuts with a donut cutter or round cookie cutter. Place on a parchment-lined tray and cover lightly. Let rise until puffy, about 20-30 minutes.
Heat oil in a deep fryer or large pot to 360°F. Fry the donuts until golden brown on both sides. Drain on paper towels.
In a saucepan, whisk together milk, cornstarch, sugar, and egg yolks. Cook over medium heat, stirring constantly, until thickened. Remove from heat, stir in vanilla extract and butter. Cover with plastic wrap directly on the surface and chill for at least 30 minutes.
Place chocolate chips in a bowl. Heat heavy cream until just boiling and pour over chocolate. Let sit for one minute, then stir until smooth.
Fill a pastry bag with the chilled pastry cream. Inject each donut with cream. Dip the top of each donut in the chocolate glaze. Allow to set before serving.