Preheat oven to 375°, place racks on upper and lower third of oven.

Line 2 baking sheets with parchment paper.

Whisk flour, crushed graham crackers, baking soda and salt in a bowl and set aside.

Cook ½ cup (1 stick) butter in a large saucepan over medium heat, swirling often and scraping bottom of pan with a heatproof rubber spatula, until butter foams, then browns, about 4 minutes. Transfer butter to a large heatproof bowl and let cool 1 minute. Cut remaining ¼ cup (½ stick) butter into small pieces and add to brown butter.

Once butter is melted, add both sugars and whisk, breaking up any clumps, until sugar is incorporated and no lumps remain. Add egg and egg yolks and whisk until sugar dissolves and mixture is smooth, about 30 seconds. Whisk in vanilla.

Fold dry ingredients into butter mixture just until no dry spots remain, fold in chocolate chips.

Using a 1½-oz. scoop form 16 balls of dough and divide between baking sheets, spacing 2” apart.

Bake cookies, rotating sheets if cookies are browning very unevenly (otherwise, just leave them alone), until deep golden brown and firm around the edges, 8–10 minutes.

Take cookie sheet out of oven and quickly press four marshmallows and three chocolate pieces into each cookie.

Return to oven and bake for an additional two minutes or until marshmallows are brown.

Remove and let cool on baking sheets.