Preheat the oven to 350 degrees F. With an electric mixer, cream the butter, sugar, vanilla, and salt until smooth. Scrape the bowl with a spatula. Turn the mixer on low and add the flour a little at a time, until it is just combined.
Line your cookie sheets with parchment paper. Roll the dough into 1-ounce balls—about 1 1/2 tablespoons. Place them on the cookie sheets and make a deep indentation in each cookie, with your thumb. Press any cracks back into place.
Bake for approximately 20 minutes. It will look like the craters have disappeared, but when the cookies cool, they’ll be back!
Scoop the Dulce de Leche into a quart-size freezer bag. Microwave it for 30-60 seconds until thin, like melted peanut butter. The bag may be hot, so wrap it in a small towel. Snip a hole in the corner of the bag and squeeze about 1 1/2 teaspoons of caramel into the centers of 24 cookies. Tap the cookies gently to help the caramel fill in the entire indention.
While the caramel is still a little tacky, sprinkle each cookie with course sea salt. Repeat this process with melted Nutella and the remaining 24 cookies.
Allow the Dulce de Leche and Nutella to cool for a little while before serving, so the centers can set.
You can put the cookies in the fridge for a few minutes to speed up the setting process of the Dulce de Leche and Nutella! Keep cooled cookies stored at room temperature in an airtight container for 4-5 days, or in the freezer for up to three months.