Preheat the oven to 325°F (163°C). Line two baking sheets with parchment paper.
In a large mixing bowl, beat the softened unsalted butter and sifted powdered sugar together using an electric mixer on medium speed until light and fluffy, about 3 to 4 minutes.
Add the vanilla extract and salt to the butter mixture and mix until combined.
Add the ube halaya to the bowl and beat until fully incorporated and the mixture is smooth and evenly purple.
Gradually add the all-purpose flour to the wet ingredients, mixing on low speed until the dough just comes together. Do not overmix.
Turn the dough out onto a lightly floured surface and knead gently to bring it together into a smooth ball.
Roll the dough out to about 1/4 inch (6 mm) thickness. Use a cookie cutter to cut out desired shapes and place them on the prepared baking sheets about 1 inch apart.
Bake the cookies in the preheated oven for 18 to 20 minutes, or until the edges are lightly golden but the centers remain pale.
Remove the cookies from the oven and allow them to cool on the baking sheets for 5 minutes before transferring to a wire rack to cool completely.
Store the cooled cookies in an airtight container at room temperature for up to 1 week.