Dissolve yeast and 1 teaspoon sugar in warm water. Stir until well combined. Let sit for 5-7 minutes to activate yeast. Mixture should foam and smell like beer.

In the bowl of a stand mixer, whisk together activated yeast mixture, remaining sugar, coconut milk, and egg.

Whisk salt into flour. Add half of flour mixture to mixing bowl. Use dough hook attachment to combine ingredients. Knead on low speed while adding softened butter one tablespoon at a time. Add remaining flour and mix on low speed until dough comes together.

Increase to medium speed. Knead until dough is smooth and clears the sides of the bowl, about 5 minutes.

Transfer dough to a lightly greased bowl. Cover bowl with reusable silicone lid (or with plastic wrap). Keep at room temperature until double has doubled in volume, about 1 hour.

Punch dough risen dough. Transfer dough to a clean working surface. Roll dough into a 20x14-inch rectangle.

If desired, mix optional ube extract into ube halaya.

Spread ube halaya evenly all over the rolled dough.

Starting at the long side, roll dough into a log. Slice log into 12 equal parts. Place rolls into a greased 9x13-inch baking pan (quarter sheet pan size).

Cover with reusable lid or plastic wrap. Let rest at room temperature until rolls have risen and puffed up, about 45 minutes.

Preheat oven to 350°F. Remove cover or plastic wrap from rolls. Pour reserved coconut milk over roll. Bake for 30-35 minutes, rotating pan midway through baking. Bake until rolls are golden brown on top and the internal temperature of the bread rolls registers at least 185°F. Let rolls rest for 10 minutes before topping with frosting.

Beat together cream cheese, butter, and ube halaya until smooth. Add powdered sugar, coconut milk, and ube extract. Beat until smooth and well combined.

Spread frosting over warm rolls.