Preheat oven to 375°F with oven racks in the top third and bottom third of oven. Pulse coconut in a food processor until finely chopped, about 10 times. Transfer to a shallow dish or bowl.

Beat butter and both sugars with an electric mixer on medium speed until smooth, about 1 minute. Add egg yolk and vanilla, and beat on low speed just until incorporated. Whisk together flour, salt, and baking powder in a small bowl, and add to butter mixture, beating on low speed just until incorporated.

Drop dough by tablespoonfuls into chopped coconut, and roll lightly to coat. Place cookies 1 inch apart on parchment paper-lined baking sheets. Press your thumb or the end of a wooden spoon into each ball, forming an indentation.

Bake in preheated oven until cookies are set and coconut is browned, about 11 minutes, switching pans top rack to bottom rack halfway through baking. Transfer baking sheets to wire racks, and immediately reshape indentations by pressing again with your thumb or spoon. Cool cookies completely, about 30 minutes.

Fill each indentation with about 1/2 teaspoon of the dulce de leche. Sprinkle cookies evenly with sea salt.