Preheat oven to 350 degrees and line baking sheets with parchment paper.
In a mixing bowl, cream peanut butter, white sugar, brown sugar, baking soda, and salt until smooth.
Add egg and vanilla and mix well.
Scoop dough using a Tablespoon measure and place on the prepared cookie sheets spacing them 2 inches apart.
Using the back of a fork, gently press down on the tops of the cookies to flatten them; do both directions.
Lightly sprinkle the tops of the cookies with granulated sugar (optional).
Bake for 10-12 minutes, rotate trays halfway through baking.
Let cookies cool on the baking sheet for 5 minutes before moving to a wire rack to cool completely. (Be careful, the cookies will be very delicate until fully cool)