Preheat oven to 350 degrees F. Line a large cookie sheet with parchment paper.
Using an electric mixer, beat together butter and sugar on medium speed until light and fluffy, about 4 to 5 minutes.
Add the egg, vanilla extract and salt until combined. Lower the speed of the mixer to low and add all-purpose flour in two additions.
Form the dough into 12 balls. Place the balls of dough on prepared baking sheet, about 1 inch apart. Using the round part of a ¼-teaspoon measure, make a deep indentation in each ball.
Bake the cookies until firm to the touch and golden on the bottom, about 10 minutes.
Allow the cookies to cool on the baking sheets for 5 minutes, then transfer and cool on wire cooling rack.
When ready to serve, fill the indentations with dulce de leche or arequipe.