In a saucepan, melt butter over medium heat, stirring occasionally until butter has browned and has a nutty scent. Remove from heat, set aside and allow to cool completely.
In a large bowl, sift together all-purpose flour, baking soda and salt.
Once brown butter has completely cooled, combine with eggs, brown sugar and granulated sugar and vanilla in a large bowl. Beat with an electric mixer until smooth.
Add flour mixture to butter mixture and beat until combined.Fold in chopped milk chocolate and graham crackers.
Cover and refrigerate for 2 hours.
Once cookie dough has been chilled, preheat over to 375º. Line a baking sheet with parchment paper and scoop dough into 2" balls, placed approx. 2" apart.
Bake for 10 minutes. Remove from heat, add 1 marshmallow to the top of each cookie, then return to oven for 2-3 minutes or until the edges are golden brown.