Melt the butter in a microwave-safe bowl in the microwave. Set aside to cool to room temperature, about 5 minutes. (Hot butter will melt the sugars and cause greasy cookies.) Once butter is cooled, scrape every bit of it into a large bowl. Add in brown and granulated sugar. Whisk briskly until completely incorporated, smooth, and combined, about 1–2 minutes.
Add in 1 full egg and 1 egg yolk (discard the whites or save for another recipe). Add vanilla. Stir until smooth.
In another bowl, stir together the flour, baking soda, and salt. Add 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 to combine.
Mix the wet and dry ingredients until just combined, avoiding overmixing. Add in white chocolate chips and stir to incorporate.
Cover the bowl tightly and chill the dough in the fridge for 1 hour and up to 8 hours.
Using a tablespoon measuring spoon, form dough balls 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.
Preheat oven to 325°F and line a sheet pan with parchment paper or a silicone 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 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—the cookies harden fast. This will give each cookie a great crispy edge.
After pressing in the edges, press a few extra 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 to a cooling rack.
Video
Notes
Note 1: Use very fresh and soft brown sugar without clumps.Note 2: Ensure baking soda is fresh before using.Note 3: Blend graham crackers in a high-powered blender to get fine crumbs—reblend any larger chunks. Spoon the crumbs into a measuring cup and level the top to get the correct measurement.Storage: Store leftover baked cookies in an airtight container at room temperature for up to a week. For dough balls, freeze them on a baking sheet, then transfer to a bag or container. Bake straight from frozen, adding a few extra minutes to the baking time.