For the cookie: Preheat the oven to 175°C. Line a rimmed baking sheet with parchment paper and spray with nonstick cooking spray.
Pulse the oats in a food processor or mini chopper until finely ground. Whisk the ground oats, flour, baking soda and salt in a medium bowl.
Mix the butter and 1 cup of the granulated sugar on medium-high speed until light and fluffy, using either a hand mixer with a large bowl or a stand mixer with the paddle attachment, about 4 minutes. Add the egg and beat until well incorporated. Beat in the peanut butter and vanilla until smooth and creamy, about 2 minutes. With the mixer on low, gradually add the flour mixture until the dough is well combined.
Scrape the dough onto the prepared baking sheet. Using your hands, flatten and form the dough into a football shape, about 1/2 inch thick. Sprinkle the remaining 2 teaspoons sugar over the surface. Bake until the cookie is light golden brown and set, 20 to 25 minutes. Allow to cool completely, about 15 minutes.
For the frosting: Beat the butter until creamy on medium speed in a medium bowl. Add the icing sugar, milk, vanilla and salt. Beat until thick and creamy. Spoon the frosting into a quart-size zippered bag. Snip a small piece from the corner and pipe to create football laces or any other desired decorations.