In a large bowl, cream together the peanut butter, butter and sugars using a hand mixer (or a stand mixer), until soft and creamy.
Add the egg and vanilla extract and continue to mix until fully combined.
In a separate bowl, whisk together the flour, baking soda and salt. Add these dry ingredients to your large bowl slowly, continuously mixing using your mixer.
Roll the dough into 1 to 2-inch balls and place on an ungreased, parchment-lined baking sheet. Using a fork, press down on each ball of dough to slightly flatten and create a criss-cross pattern.
Bake for 8-10 minutes or as soon as tops begin to slightly crack. To keep them soft and chewy, be careful to not overbake. Let cool on the baking sheet for a few minutes and then transfer to a wire rack.