Make the custard: Heat 1 1/2 cups milk and vanilla in a saucepan until simmering. In a bowl, whisk egg yolks, 1/3 cup sugar, and cornstarch until pale.

Slowly whisk half the hot milk into the egg mixture to temper. Return everything to the saucepan and cook over medium heat, whisking constantly, until thick and boiling for 1 minute.

Remove from heat, whisk in 1 tbsp cold butter. Strain through a fine mesh sieve into a bowl, cover with plastic wrap touching the surface, and chill for at least 3 hours.

Make the dough: In a stand mixer, combine bread flour, yeast, 1/4 cup sugar, and salt. Add room temperature eggs and warm milk. Mix with a dough hook until a shaggy dough forms.

With the mixer on low, gradually add softened butter one tablespoon at a time until incorporated. Knead on medium speed for 10-15 minutes until the dough passes the windowpane test.

Place dough in a greased bowl, cover, and let rise in a warm place for 1-1.5 hours until doubled. (For best results, chill overnight after this step).

Divide dough into 8 equal pieces. Roll each into an oval bun shape. Place on a parchment-lined baking sheet, cover, and let rise for 1 hour until puffy.

Preheat oven to 350°F (175°C). Brush buns with egg wash and bake for 15-18 minutes until golden brown. Cool completely on a wire rack.

Whisk the chilled custard to smooth it out. Fold in half of the mini chocolate chips and transfer to a piping bag.

Slice a slit down the top of each cooled bun. Pipe the custard generously inside and sprinkle with remaining mini chocolate chips.