Preheat the oven to 356º F (180° C). Line two large rectangular baking sheets with parchment paper. In a medium bowl, sift the flour with the baking soda and salt.
Separately, in a large bowl add the butter with the two types of sugar and cream with an electric mixer or hand whisk for 2 minutes until light and fluffy.
Add in the peanut butter and beat until blended. Add the egg with the vanilla extract and beat for 1 more minute until smooth.
Add the flour mixture in two separate additions, integrating with a rubber spatula until you get a very smooth dough.
Using your hands, portion the dough into 24 balls about the size of 2 tablespoons each (2.5 cm wide) and place them on the baking sheets, at least 5 cm apart from each other as they will spread a lot during baking.
With a fork, lightly flatten each ball of dough twice, forming a criss-cross pattern on the cookies. If the dough starts to stick to the fork, roll it in a little white sugar.
Bake the cookies for 10-12 minutes until they are firm and lightly browned on the edges but not quite in the center. Remove from the oven and let them cool down completely on a wire rack. The cookies can be stored in a well-sealed tupperware container for up to 1 week.