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

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

In the bowl of a stand mixer fitted with the paddle attachment, 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 until fully incorporated. Stir in the vanilla extract until just combined.

Reduce the mixer speed to low. Gradually add the prepared dry ingredient mixture to the wet ingredients. Mix only until the flour streaks have almost disappeared. Avoid overmixing the dough.

Remove the bowl from the mixer and use a sturdy spatula. Gently fold the 2 1/2 cups of rolled oats into the cookie dough until they are evenly distributed.

Use a medium cookie scoop (about 1 1/2 tablespoons) to portion the dough. Place the dough balls onto the prepared baking sheets, spacing them about 2 inches apart. Use the tines of a fork to lightly press down on the top of each dough ball in a cross-hatch pattern.

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 removed from the oven.

Allow the cookies to cool on the baking sheet for 5 minutes before moving them. Transfer the cookies using a spatula to a wire rack to cool completely.