In a large mixing bowl, combine flour and sugar. Cut in the chilled butter using a pastry cutter or your fingers until the mixture resembles coarse crumbs.
Add the egg and milk. Mix until the dough comes together. If the dough is too dry, add a little more milk, one tablespoon at a time.
Wrap the dough in plastic wrap and refrigerate for at least 30 minutes.
In a small saucepan, combine the sweetened condensed milk, evaporated milk, and vanilla extract. Mix well.
In a small bowl, dissolve cornstarch in water and then add it to the milk mixture.
Cook over medium heat, stirring constantly until the mixture thickens (about 5-7 minutes). Remove from heat and let it cool.
Preheat your oven to 375°F (190°C) and line a baking sheet with parchment paper.
On a floured surface, roll out the dough to about 1/8 inch thickness. Use a round cutter (about 3-4 inches in diameter) to cut out circles.
Place a small spoonful of the filling in the center of each dough circle. Fold the dough over to form a half-moon shape and press the edges together to seal. You can use a fork to crimp the edges for a decorative touch.
Place the empanadas on the prepared baking sheet.
Bake in the preheated oven for 15-20 minutes, or until the empanadas are golden brown.
Remove from the oven and let them cool slightly before serving. Optionally, sprinkle with ground cinnamon for extra flavor.
Enjoy your meal!