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Whisk the flour, baking powder, baking soda, and salt together in a medium bowl. Set aside.
Using a hand mixer or a stand mixer fitted with a paddle attachment, cream the butter and both sugars together on medium speed until smooth, about 1-2 minutes. Add the eggs and beat on high until combined, about 1 minute. Scrape down the sides and bottom of the bowl as needed. Add the peanut butter and vanilla, then beat on high speed until combined.
Add the dry ingredients to the wet ingredients, then mix on low until combined. With the mixer running on low speed, add the peanuts, if using. Dough will be thick and soft.
Cover and chill the dough for 1 hour in the refrigerator (and up to 2-3 days). If chilling for longer than a few hours, though, allow to sit at room temperature for at least 30 minutes before rolling and baking because the dough will be quite hard.
Preheat oven to 350°F (177°C). Line 2-3 large baking sheets with parchment paper or silicone baking mats. (Always recommended for cookies.) Set aside.
Roll cookie dough into balls, about 1.5 Tablespoons of dough per cookie, and then roll the balls in granulated sugar. Use a fork to make a crisscross indent on top of each. Bake each batch for 10-12 minutes until very lightly browned on the sides. The centers will look very soft.
Remove from the oven. Cool cookies on the baking sheets for 5 minutes before transferring to a wire rack to cool completely.
Cookies stay fresh covered at room temperature for up to 1 week.