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, and mix until smooth. Add in the flour mixture, and mix until fully combined. Roll the cookie dough into balls, and flatten each one slightly.
Place one Oreo cookie onto each piece of dough, form the dough into a ball around the Oreo cookie. Place onto a large ungreased baking sheet. Bake for 10-12 minutes.