Preheat the oven to 350 F. In a medium bowl, whisk together flour, salt, and baking soda. In a large bowl, combine butter and sugar until fluffy.
Add in the egg and vanilla extract, mix until smooth. Add in the flour mixture, and mix until the dough is fully combined. Roll the cookie dough into balls, and place one unwrapped peppermint patty onto each piece of dough.
Roll into a ball, and place the cookies onto a large ungreased baking sheet. Bake for 10-12 minutes.