In a medium bowl, whisk the melted butter, brown sugar, and granulated sugar together until no brown sugar lumps remain. Whisk in the egg, then the egg yolk. Whisk in the vanilla extract. Pour the wet ingredients into the dry ingredients and mix together with a large spoon or rubber spatula. The dough will be very soft, yet thick. Fold in the chocolate chips. They may not stick to the dough because of the melted butter, but do your best to combine them. Cover the dough and chill in the refrigerator for 2-3 hours or up to 3-4 days. Chilling is mandatory. I highly recommend chilling the cookie dough overnight for less spreading.
Preheat oven to 350 degrees.
Grease the bottom of your skillet with olive oil. You don't need much, a tablespoon is plenty. Use a paper towel to spread the oil on the bottom and sides of the skillet. Add your chocolate chip cookie batter to the skillet and spread evenly.
Bake at 350 degrees for 30-35 minutes or until golden brown. Use a toothpick to make sure the center is done to your liking. Serve warm.
Make ahead tip: You can make the cookie dough and chill it in the refrigerator for up to 2-3 days. Allow to come to room temperature then continue with step 4.
When baked the Skillet Cookie stays fresh in a air tight container at room temperature for up to 1 week.