Bloom the yeast: While the halaya cools, whisk the brown sugar, coconut milk and yeast in the bowl of your stand mixer. Cover and let the yeast activate for 15 minutes until fragrant and foamy.
7 g yeast, 28 g brown sugar, 169 g coconut milk
Make the dough: Add the remaining ingredients, including the cooled ube halaya, to the yeast mixture. Mix on low speed using the dough hook until a shaggy dough forms. Gradually increase the speed to medium and continue mixing until the dough is smooth and pliable.
113 g sourdough discard, 480 g all-purpose flour, 10 g ube extract, 9 g salt
Proof the dough: Round the dough into a tight ball and transfer it to a lightly-oiled bowl. Cover with plastic wrap and let the dough rise until doubled in volume, about 45 minutes to 1 hour, depending on the temperature of your kitchen.
Prep: Line a large metal baking pan (9 by 13 inches) with parchment paper.
Shape the dough: Divide the dough into 12 portions. Round each piece of dough into a tight ball. Turn the dough over and pinch to seal the seam. Arrange the dough into the prepared pan.
Proof the rolls: Cover the pan with plastic wrap and let the dough rise until puffy and almost touching each other.
While the rolls proof, preheat the oven to 350 F.
Brush with egg wash: Whisk the egg to break up the yolk. Brush the top of the rolls with egg wash using a pastry brush.
Bake the rolls: Bake the rolls for 25 to 30 minutes or until the tops are browned and the center of the rolls measure at least 190 F when probed with an instant-read thermometer.
1 egg