Preheat the oven to 375°F. Line baking sheets with parchment paper.
In a medium saucepan, melt 1/2 cup butter over medium-high heat. Cook and stir for about 3 minutes until butter has browned, then transfer to a mixing bowl and add remaining butter. Stir until melted, then set aside to cool for 10 minutes.
¾ cup salted butter
Add brown sugar to the cooled butter. Mix until combined. Add eggs and vanilla, mixing well.
1 ½ cups (300g) light brown sugar, 2 large eggs, 1 teaspoon vanilla extract
Add flour, baking soda, baking powder, and salt. Mix just until combined. Cover dough with plastic wrap and chill for 1 hour.
2 ¼ cups (317g) all-purpose flour, ½ teaspoon baking soda, ¼ teaspoon baking powder, ½ teaspoon salt
Scoop 1 tablespoon of cookie dough and shape into a small ball. Place on baking sheets, 3 inches apart, and form an indentation in the top with a finger or backside of a measuring spoon.
Fill each indentation with a small amount of dulce de leche, about 1/2 teaspoon.
½ cup dulce de leche
Shape another tablespoon of cookie dough into a ball and press an indentation in it as well, then turn it over and use it to cover the dulce de leche, pinching the edges together to seal dulce de leche inside. Repeat until all dough has been used.
Bake for 11-13 minutes until barely golden brown, then remove from the oven and cool for 5 minutes before cooling completely on a wire rack.