In a small bowl, combine 1 cup of warm milk, 2 ½ teaspoons of active dry yeast, and 1 tablespoon of sugar. Let it sit for about 10 minutes until frothy.

In a large mixing bowl, whisk together 1 large egg, ½ cup of vegetable oil, ½ cup of ube halaya, and 1 teaspoon of ube extract until smooth.

Add the yeast mixture to the wet ingredients. Gradually incorporate 3 ½ cups of all-purpose flour, ½ cup plus 1 tablespoon of sugar, and 1 teaspoon of salt. Stir until a soft dough forms.

Transfer the dough to a floured surface and knead for 8 to 10 minutes until smooth and elastic.

Place the kneaded dough in a greased bowl, cover with a towel, and let rise for about 1 hour until doubled in size.

Punch down the dough gently, divide into 16 pieces, flatten each, place a cube of cheese inside, and seal tightly.

Roll each filled ball in breadcrumbs and arrange on a parchment-lined tray for the second rise.

Cover with a towel and let rise for 30 to 40 minutes until puffy.

Preheat oven to 350°F. Bake rolls for 18 to 20 minutes until golden brown.

Cool rolls on a wire rack for about 10 minutes before serving.