Whisk the flour, baking soda, and salt together in a medium bowl. Set aside.

Using a hand mixer or stand mixer fitted with paddle attachment, beat the butter for 1 minute on high speed until creamy. Change to medium-high speed and beat in the brown sugar and ¼ cup granulated sugar until completely creamed, about 2 minutes. Add the egg and beat until combined, about 1 minute. Scrape down the bowl as needed. Add the peanut butter and vanilla extract and beat until combined.

Add the dry ingredients to the wet ingredients and beat on low speed until combined. Cover and chill the cookie dough in the refrigerator for at least 1–2 hours, and up to 3 days.

Preheat oven to 350°F (177°C). Line baking sheets with parchment paper or silicone mats.

Roll chilled cookie dough into balls, about 1 scant Tablespoon of dough each. Roll each ball in sugar and arrange on baking sheets about 2 inches apart. Use a fork to make a criss-cross indent on top of each and reshape into a thicker disc.

Bake for 11-12 minutes until edges appear set and are lightly browned. The centers will look soft.

Allow cookies to cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.

Cookies stay fresh covered at room temperature for up to 1 week.