Preheat the oven to 325°F (163°C). Line a baking sheet with parchment paper.
In a large mixing bowl, cream together the softened unsalted butter and granulated sugar until light and fluffy, about 3-4 minutes using an electric mixer on medium speed.
Add the ube halaya and vanilla extract to the butter and sugar mixture. Mix until fully incorporated and smooth.
In a separate bowl, whisk together the all-purpose flour, salt, and cornstarch.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Do not overmix.
Turn the dough out onto a lightly floured surface and gently knead a few times to bring it together. Flatten the dough into a disk and wrap it in plastic wrap. Chill in the refrigerator for 30 minutes.
Remove the dough from the refrigerator and roll it out to about 1/4-inch thickness on a lightly floured surface.
Use a cookie cutter or knife to cut the dough into desired shapes (about 2 inches in diameter). Place the cookies on the prepared baking sheet about 1 inch apart.
Bake in the preheated oven for 18-20 minutes, or until the edges are lightly golden but the cookies remain pale.
Remove from oven and allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.