In a large mixing bowl, cream together the softened butter and sugars. Then, stir in the vanilla extract, peanut butter and egg until just combined. Add the flour, 1/2 teaspoon salt, and chopped peanuts and stir until dough forms and no loose flour remains.
Spoon onto a prepared baking sheet covered with parchment paper or a baking mat. Sprinkle sea salt over the tops of the cookies.