In a medium saucepan, whisk together the eggs, sugar, cornstarch, and cake flour until smooth. Heat the milk in another pan until gently simmering and temper the egg mixture with hot milk while whisking. Return to heat and cook until thickened for about 5 minutes. Stir in butter and vanilla, then chill.

In a stand mixer, combine bread flour, cake flour, sugar, condensed milk, water, cold milk, half of the whisked egg, and yeast. Knead until smooth, let rest for 30 minutes, then add sea salt and softened butter, knead until silky.

Shape the rested dough into a ball, place in a greased bowl, and cover. Let rise in a warm place for 1 to 1.5 hours until doubled in size.

After rising, punch down the dough, divide into 8 portions, and shape into buns while reserving some center for custard. Cover and let rest for 10 minutes.

Place shaped buns on a lined baking sheet with space in between, cover, and let rise for an additional hour until puffed.

Preheat oven to 375°F (190°C). Brush tops with whisked egg, optionally top with custard or pearl sugar, and bake for about 12 minutes until golden brown. Cool on a wire rack before serving.