Preheat the oven to 350 degrees fahrenheit (177 celsius). Grease an 8x8 inch square baking pan with oil/butter or line with parchment paper.
In a large bowl using a hand mixer (or use a stand mixer), cream the butter, cream cheese, and sugar together. Add the egg and vanilla and beat until fluffy. Add the salt, baking soda, and half of the GF flour and beat until combined. Scrape the sides of the bowl with a spatula. Add the rest of the gluten free flour and mix until smooth and fluffy.
Spoon half of the batter into the prepared baking pan and spread it out evenly. Place four rows of four Oreos across the entire pan (16 Oreos total), to create a layer of the cookies. Spread the rest of the batter on top of the cookies, sprinkle the crushed Oreos over the surface, and bake for 25-35 minutes or until the surface is golden brown. Remove from the oven and let them cool for at least 15 minutes. Slice into bars, serve, and enjoy!