Preheat the oven to 350 F. In a medium bowl, whisk together flour, salt, and baking soda. In a large bowl, combine butter with peanut butter and both sugars until fluffy. Add in the egg and vanilla extract, and mix until smooth.
Add in the flour mixture, mix until it is fully combined. Roll the cookie dough into oval shapes and flatten slightly. Place the cookies onto a large ungreased baking sheet at least 2-inches apart.
Bake for 10-12 minutes. Remove from the oven and let cool completely. Pipe icing onto each cookie in a football pattern.