Mix flour, baking soda, salt and graham crackers together and set aside.
In a large skillet, start melting the butter. Once the butter is fully melted and you can 'hear' it. Take it off the heat and place it in your mixing bowl.
Fill another bowl with ice water and place the bottom of the bowl (the one that has melted butter in it) in the ice water. I know this sounds extra, but this helps the butter get back to room temp, but stay in a liquid state.
Once the butter has cooled, start adding brown sugar, granulated sugar, sourdough discard, vanilla extract, lime juice, lime zest, and the egg. Mix until everything is well incorporated and smooth.
Once the wet ingredients are mixed well, keep your mixer on a low setting and start gradually adding the dry ingredients.
After the dough has come together, it might look a bit like cake batter. If you want to thicken it a bit, add more flour.
Line your cookie sheet with parchment paper, and take a medium sized ice cream scoop to the dough.
Once your cookie sheet is filled, chill them in the fridge for a minimum of 15-20 minutes before baking.
While your cookies are chilling, preheat your oven to 350 degrees.
When your cookies are done chilling, add extra crushed graham crackers on top, and bake your cookies for 13-14 minutes.
While your cookies are baking, add powdered sugar, vanilla extract, and lime juice to a bowl. Start gradually adding water, milk, or heavy cream to the mixture until you have your desired consistency.
Let the cookies cool for 5 - 10 minutes before drizzling icing on top.
After you add the icing, top with more graham crackers and enjoy!