In a saucepan, combine 1/2 cup of unsalted butter, 1 cup of whole milk, 1/2 cup of water, 1 tablespoon of sugar, and a pinch of salt. Bring the mixture to a rolling boil over medium heat, then quickly stir in 1 cup of all-purpose flour. Keep stirring until the mixture forms a smooth ball that pulls away from the sides of the pan, which should take about 2-3 minutes.

Remove the choux pastry dough from heat and allow it to cool for 5-10 minutes. Once it feels warm to the touch, add in 4 large eggs, one at a time, mixing well with a wooden spoon after each addition. The dough should become glossy and smooth.

Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper and use a piping bag to form 2-inch rounds of choux pastry on the sheet, spacing them about 2 inches apart. Bake in the preheated oven for 10 minutes, then reduce the temperature to 350°F (175°C) and bake for an additional 20 minutes, or until the pastries are golden brown and puffed up.

While the pastry cools, it's time to prepare the delightful ube filling. In a saucepan over medium heat, whisk together 1/2 cup of ube halaya, 1 cup of milk, 1/2 cup of heavy cream, 1/4 cup of sugar, 2 tablespoons of cornstarch, and 3 egg yolks. Stir continuously until the mixture thickens, which should take roughly 5-7 minutes. Once thickened, remove from heat and stir in 1 teaspoon of vanilla extract, then let cool.

Once the choux pastries are completely cool, carefully poke a small hole in the side of each puff with a skewer or a knife. Use a piping bag to fill each pastry with the rich ube cream, filling them generously. Finally, if desired, dust the filled Ube Cream Puffs with powdered sugar for an extra touch of sweetness before serving.