In a stand mixer bowl with paddle attachment, cream together softened butter and sugar until smooth.
Add the egg, egg yolk, and vanilla extract, mixing until combined.
Gradually add half of the flour and mix on low speed until combined, then add the remaining flour and mix until incorporated.
Cover the dough and refrigerate for 1 hour.
Preheat the oven to 375°F and line two baking sheets with parchment paper.
Shape the dough into rounded 1 tablespoon balls and place them 2 inches apart on the prepared baking sheets.
Flatten each ball slightly with your palm and use a teaspoon to create an indentation in the center of each cookie.
Bake for 10 to 12 minutes, until the edges start to brown lightly.
Remove from oven and if needed, press down the indentations again while warm; cool on a wire rack to room temperature.
Fill each indentation with dulce de leche using a small spoon or pastry bag, then sprinkle with sea salt flakes.
Enjoy immediately or store as directed.