In a small saucepan, whisk flour into milk and heat, stirring constantly, until it thickens. It will be the consistency of a thick paste.*
Remove from heat, and let it cool to room temperature. This step is key. If the milk mixture is at all warm, it won't come together properly.
Stir in vanilla.
Once the mixture is cool, cream the butter, sugar, and salt together until light and fluffy on medium-high, about 5 minutes.
Then add the completely cooled milk mixture.
Beat it for about 5 minutes on medium-high to high until it looks like whipped cream. It may look separated at first, so keep beating it until it comes together and looks like whipped cream.
Fold in cookie crumbs.
Frost cake or cupcakes of your choice.
Notes
*The cooked flour mixture continues to thicken as it sits. So, if you think it might still be a little too wet, try letting it sit longer before continuing with the recipe.**If you are going to pipe the frosting, you'll want the cookie to be crushed to crumbs. Otherwise, the cookie pieces will get stuck in the piping tips. If you aren't going to pipe the frosting, feel free to leave the cookies in a little larger pieces. I've made it both ways, and either way it delicious.***Check out all my tips for making this frosting just above the recipe.****Nutrition values are estimates.