In a medium mixing bowl, combine the flour, baking soda, and salt, and set aside.
In a large mixing bowl, using a handheld or stand mixer fitted with a paddle or whisk attachment, cream the butter, sugar, and brown sugar on medium-high speed for 2-3 minutes. Add the egg and vanilla and mix for about 1 minute, until well combined and creamy. Mix in the peanut butter for about 30-60 seconds until combined. Then add the flour mixture and mix on low speed just until incorporated, about 30-60 seconds. Stir in the oats just until combined. Then add the chocolate chips, if used, and mix just until well distributed. The dough will be slightly sticky (depending on the absorption rate of the flour). Cover and chill for 1 hour.
Line two baking sheets with parchment paper and preheat the oven to 350°F (175°C).
Scoop 16 equal-sized balls (about 2 tablespoons [50g]) of cookie dough, roll into even balls, and place them 3 inches (7.5 cm) apart, on the prepared baking sheets and flatten with your fingers that they are about ¾ inch (1.9cm) thick. Bake one sheet at a time for 10-12 minutes or until the edges are set and lightly browned and the centers look undone, pale, puffy, and dry.
Remove from the oven and let cool on the baking sheet for about 5 minutes, or until firm enough to move. They will continue baking on the baking sheets. Then, transfer to a wire rack and let cool completely. The cookies will stay fresh in an airtight container at room temperature for up to 3 days or freeze for up to 3 months.