In a large mixing bowl, dissolve the sugar and yeast in warm milk. Let the mixture stand for about 5 minutes, until foamy.
Combine flour and salt. Add the mixture gradually to the yeast mixture while stirring. Continue to stir until a dough forms and gathers at the center of the bowl.
Turn the dough over on a lightly floured board. Knead it until it is smooth. Shape it into a ball and place it in a bowl. Cover and let it rest for 10 minutes.
Roll the dough into a 12x8 inch rectangle. Dot dollops of softened butter in a single layer all over the dough. Fold the upper third of the dough over to the center, then fold the lower third up to create a business letter style fold. Wrap the dough in plastic and chill it in the freezer for 10 minutes.
Remove the dough from the freezer and unwrap. With the short edge close to you, roll it once more to a 12x8 inch rectangle. Fold it again business letter style. Wrap it in plastic and chill in the freezer for another ten minutes.
Finally, remove the dough from the freezer and unwrap it. With the short edge close to you, roll the dough into an 8x14 inch rectangle. Spread the ube jam (ube halaya) all over the dough.
Roll the long side of the dough towards the center to form a tight log up to the other side. Slice the dough into 14-16 one inch thick portions. Place each portion on greased 12-cup muffin pans. Cover the pan loosely with a clean kitchen towel. Let it rise for 45 minutes, or until puffy.
Preheat oven to 375 F. Bake the buns at 375 F for 15-18 minutes or until lightly golden. Let cool slightly until easy to handle. Toss in sugar to coat. Top with shredded cheese.