Preheat the oven to 325°F (163°C). Line two baking sheets with parchment paper.
In a large mixing bowl, cream the softened unsalted butter and sifted powdered sugar together until light and fluffy, about 3-4 minutes using an electric mixer on medium speed.
Add the vanilla extract and salt to the butter mixture and beat until combined.
In a separate bowl, whisk together the all-purpose flour and cornstarch.
Gradually add the flour mixture to the butter mixture, mixing on low speed until just combined.
Add the ube halaya to the dough and gently fold it in with a spatula until the dough is evenly colored and combined. Do not overmix.
Turn the dough onto a lightly floured surface and knead gently just to bring it together.
Roll the dough out to about 1/4 inch (0.6 cm) thickness.
Use a cookie cutter to cut out desired shapes and place them on the prepared baking sheets about 1 inch apart.
Bake in the preheated oven for 18-20 minutes, or until the edges are just beginning to turn golden.
Remove the cookies from the oven and allow them to cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.