Preheat your oven to 350°F (175°C). Line two or three large baking sheets with parchment paper or silicone baking mats and set them aside.

In a medium mixing bowl, whisk together the flour, baking soda, and salt until they are thoroughly combined. Set this dry mixture aside.

In the bowl of a stand mixer, combine the softened butter, peanut butter, brown sugar, and granulated sugar. Beat the mixture on medium speed until it is light, fluffy, and creamy (about 3 to 4 minutes). Scrape down the sides of the bowl as needed.

Add the eggs one at a time, beating well after each addition. Stir in the vanilla extract until just combined and the mixture is smooth.

Reduce the mixer speed to low. Gradually add the prepared dry ingredient mixture (flour, salt, and baking soda) to the wet ingredients. Mix only until the flour streaks have almost disappeared. Do not overmix.

Remove the bowl from the mixer. Use a sturdy spoon or spatula to gently fold the 2 1/2 cups of rolled oats into the cookie dough until they are evenly distributed throughout.

Scoop the dough using a medium cookie scoop (about 1 1/2 tablespoons) and roll into balls. Place the dough balls onto the prepared baking sheets, spacing them about 2 inches apart.

Using the tines of a fork, lightly press down on the top of each dough ball in a classic crisscross pattern. This helps them spread slightly and provides the traditional peanut butter cookie look.

Bake for 10 to 12 minutes. The edges of the cookies should be set and lightly golden brown, but the centers may still look slightly soft when you remove them from the oven.

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