Preheat the oven to 350°F and line two large baking sheets with parchment paper.

In a medium bowl, whisk together the flour, baking soda, and salt.

In the bowl of a stand mixer, beat the butter, brown sugar, and the ¼ cup granulated sugar until light and fluffy, 2 to 3 minutes. Add the peanut butter, egg, and vanilla and mix until creamy. Gradually add the dry ingredients and mix on low speed until just combined.

Scoop slightly heaping tablespoons of the dough, then use your hands to roll it into balls. Place the ½ cup granulated sugar in a small bowl and roll the cookie dough balls in it. Arrange on the prepared baking sheets, leaving at least 2 inches between balls. Press down on the tops of the balls with a fork to form a cross-hatch pattern.

Bake, one sheet at a time, for 10 to 12 minutes, 10 minutes for a softer cookie and 12 minutes for a crispier cookie. The cookies will be just set around the edges but still seem underdone in the middle. Allow to cool on the baking sheet for 10 minutes before transferring to a wire rack to cool completely.

Store in an airtight container at room temperature for up to 5 days.