Preheat the oven to 175 C / 350 F. Line a sheet pan 33 x 46 cm / 13 x 18 inch with parchment paper.
In a mixing bowl, stir the peanut butter, 1 cup granulated sugar, egg, and vanilla together until smooth and glossy.
Sprinkle the baking soda and salt over the dough and mix until fully combined. The dough will stiffen slightly as it comes together.
Scoop the dough into 26 g balls with a cookie scoop or tablespoon, rolling each one between your palms until smooth.
Roll each ball in the reserved 2 tbsp sugar to coat lightly, then place on the sheet pan 5 cm / 2 inches apart.
Press each ball flat with a fork, once vertically and once horizontally, to make the classic crisscross pattern.
Bake for 8 to 10 minutes, until the edges are set and lightly golden but the centers still look soft.
Cool the cookies on the pan for 5 minutes, they're fragile straight from the oven, then transfer to a wire rack to finish cooling.