In a stand mixer, combine bread flour, granulated sugar, and instant dry yeast; mix briefly to combine. Gradually add whole milk, egg, and melted unsalted butter, kneading on low speed until a rough dough forms.

Incorporate fine salt and knead for 15-30 minutes, until the dough is smooth and elastic, and pulls away from the bowl.

Shape the dough into a ball and place it in a greased bowl, covering with plastic wrap or a clean kitchen towel. Let it rise in a warm area for about 60 minutes, or until doubled in size.

Mix the ube jam in a bowl until spreadable, then stir in a few drops of ube extract. Shred your choice of cheese, ensuring readiness for the filling.

Punch down the dough to release air bubbles, divide into 10 pieces. Flatten each piece, spread ube jam, sprinkle cheese, then roll into a log. Chill in the freezer for about 15 minutes.

Remove the logs from the freezer, slice them open lengthwise, twist to create a braid and place on a baking tray. Cover with a towel and let rise for another 30-40 minutes, or until nearly doubled in size.

Preheat your oven to 180°C (350°F).

Prepare egg wash by whisking one egg with water. Brush over braided buns for a shiny finish and bake for 15-20 minutes until golden brown and hollow-sounding.

Allow buns to cool slightly on a wire rack before serving.