Preheat the oven to 350°F (175°C) and position one rack in the center. Line two large baking sheets with parchment paper or silicone baking mats.
In a large mixing bowl, beat the softened butter, granulated white sugar, and packed light brown sugar with an electric hand mixer or stand mixer on medium speed for 2 to 3 minutes until light, fluffy, and paler in color. Scrape down the sides with a rubber spatula.
Add the creamy peanut butter and mix on medium speed for about 1 minute until fully combined. Add the egg and vanilla, mixing on medium-low speed for 30 seconds until just incorporated. Stir the sourdough discard well, then add it and mix on low speed until fully combined. Scrape down the sides of the bowl again.
In a separate small bowl, whisk together the all-purpose flour, baking soda, baking powder, and fine salt. Add the dry ingredients to the wet ingredients all at once and mix on low speed until just combined and no streaks of dry flour remain. Fold in any remaining dry patches by hand with a rubber spatula. Do not overmix.
Pour 3 to 4 tablespoons of granulated sugar into a small shallow bowl. Using a 1.5 tablespoon cookie scoop, portion out even balls of dough, roll them into neat spheres, and coat each ball completely in the sugar. Place the sugar-coated dough balls onto the prepared baking sheets, spacing them about 2 inches (5cm) apart.
Using the back of a fork, gently press down on each dough ball to flatten into a disc about 1/2 inch (1.25cm) thick. Rotate the fork 90 degrees and press again to create the classic crosshatch pattern. If the fork sticks, dip it lightly in granulated sugar between pressings.
Bake one sheet at a time on the center rack for 10 to 12 minutes, until the edges are set and just starting to turn golden, while the centers still look slightly underdone and soft. Allow the oven to return to temperature before baking the second tray.
Remove the baking sheet from the oven and let the cookies rest on the pan for 5 minutes. Transfer them to a wire cooling rack and cool for at least 10 minutes before serving. They will firm up beautifully as they cool.