BUTTER AND SUGARS: Melt the butter in a microwave-safe bowl in the microwave. Set aside and cool to room temperature, about 5 minutes. ( If the butter is hot when it is added to the sugars, it will melt the sugars and cause greasy cookies.) Once butter is cooled, scrape every bit of it into a large bowl. Add in the brown and white sugar. Whisk briskly until completely incorporated, smooth, and combined-- about 1-2 minutes.
EGGS AND VANILLA: Add in 1 full egg and then 1 egg yolk (discard the whites or save for another recipe). Add in the vanilla. Stir until smooth.
DRY INGREDIENTS: In another bowl, stir together the flour, baking soda, and salt. Add full graham crackers to a blender or food processor and crush until you have 1 cup (117g) of very fine crumbs (gently add to the measuring cup; don't pack it in). Add the graham cracker crumbs to the dry ingredients. Stir everything to combine.
COMBINE: Mix the wet and dry ingredients until just combined, avoiding overmixing. Add in white chocolate chips and stir to incorporate.
CHILL: Cover the bowl tightly and chill the dough for 1 hour and up to 8 hours.
ROLL COOKIE DOUGH BALLS: Using a tablespoon measuring spoon, form balls of dough using 2 tablespoons of dough per cookie (45g if you have a food scale). Roll the cookie dough balls large and tall (rather than wide). Place on a parchment-paper-lined plate and chill again for 20 minutes.
BAKE: Preheat the oven to 325 degrees F and line a pan with parchment paper or a Silpat liner. Space 6 cookies per cookie sheet (to allow PLENTY of room for spread; they spread a lot). Bake for 10-14 minutes or until ever-so-slightly browned at the edges or no longer gooey looking on top. Remove from the oven and immediately use a spoon or the back of a spatula to press the edges of each cookie into the center. (See video or photos). Work quickly, because the cookies harden fast. This will give each cookie a great crispy edge.
DECORATE: After pressing in the edges, press a few additional white chocolate chips into the tops of the cookies. Coarsely break up a few additional graham crackers and press small pieces onto the tops of the cookies to decorate/garnish. Let cookies stand on the tray for 4 more minutes then use a spatula to remove the cookies from the sheet pan to a cooling rack.
Video
Notes
Note 1: Brown Sugar: Use very fresh and soft brown sugar without clumps.Note 2: Baking Soda: Ensure baking soda is fresh before using.Note 3: Graham cracker crumbs: Blend in a high-powered blender to get fine crumbs -- reblend any larger chunks. Spoon the crumbs into a measuring cup and level at the top to get the correct measurement.