Grease a 17 x 12 x 1 inch baking sheet (or line it with parchment), and preheat oven to 350 degrees F.
Whisk together flour, oats, baking soda, and salt, and set aside.
In a large mixing bowl, cream butter, granulated sugar, brown sugar, and eggs. Add peanut butter and vanilla, and beat 3 minutes, more. Mix in dry ingredients, and stir just until combined. Dump dough onto prepared baking pan, and use a spatula to spread to about 1 centimeter from the edges. Bake at 350 degrees for 20-25 minutes. Cool completely on a wire rack.
Frosting Directions:
Combine all ingredients, and beat until fluffy and light. Spread evenly over cooled cookie. Cut into rectangles measuring 4 x 2 inches.