In a medium mixing bowl, whisk together the brown sugar, white sugar, egg, vanilla extract, baking soda, and salt until well combined.
Whisk in the peanut butter until the mixture is smooth, resembling a thick brownie batter.
Chill the dough in the refrigerator for at least 1 hour until thickened.
Preheat your oven to 325°F (160°C) and line a baking sheet with parchment paper.
Scoop the chilled dough using a 1 ½ tablespoon measure and place dough balls onto the prepared baking sheet, leaving space between them.
Sprinkle the tops with cane sugar if desired. Gently flatten the tops with your fingers or a fork.
Bake for 11-15 minutes, until the edges are golden brown and the tops show slight cracks.
Allow the cookies to cool on the baking sheet for 15 minutes before transferring to a wire rack to cool completely.