Preheat oven to 350 degrees F (175 degrees C).
In a medium bowl, stir peanut butter and sugar together by hand until smooth.
Add in the eggs, one at a time.
Stir in the baking soda, salt, and vanilla.
Using a medium ice cream scoop, (roughly 1-2 tablespoons) scoop out peanut butter cookie dough and set it on top of cookie. Repeat until Oreo's all have peanut butter cookie dough on them.
Carefully press the dough around the cookie (trying not to break the cookie) so that all of the Oreo is covered in peanut butter dough.
Repeat until all of the Oreo's are covered.
Using a fork dipped in flour, press the fork into the top of the cookie and create the criss-cross pattern.
Bake for 7-9 minutes or until the cookies are just starting to brown on the edges.
Allow cookies to cool on baking sheet for 5 minutes before removing to a wire rack to cool completely.