Preheat the oven to 350°F (177ºC).
Melt the butter (1/2 cup) in a small saucepan over medium-high heat. Once the butter is completely melted, stir it occasionally while it continues to cook for about 5 minutes, or until a deep amber color develops. Let the brown butter cool in the saucepan for at least 20 minutes.
Combine the butter, brown sugar (1/2 cup), and granulated sugar (1/2 cup) in a large mixing bowl. Use an electric hand mixer to cream the ingredients together for 2-3 minutes, or until the sugar is completely dissolved and the batter is light and fluffy.
Add the egg (1) and vanilla extract (1 teaspoon), mixing just until they are incorporated.
Reduce the speed of your hand mixer and gradually add the flour (1 1/2 cups), baking powder (1/2 teaspoon), and baking soda (1/2 teaspoon). Mix only until the dry ingredients are just combined. Next, use a rubber spatula to thoroughly mix the batter, being sure to scrape the bottom and sides of the bowl to eliminate any dry pockets. Make sure that all ingredients are well combined and no dry ingredients remain. Do not overmix!
Gently fold in the chocolate chips (1 cup) and crushed graham crackers (1/2 cup), ensuring they are evenly distributed throughout the cookie dough.
Arrange balls of cookie dough onto a baking sheet and bake for 12 minutes.
As soon as the cookies come out of the oven, top them with 3-4 mini marshmallows. Set your oven to broil on low heat and put the cookies back in the oven for 1-3 minutes, or until the marshmallows are golden brown. Let the cookies cool for a few minutes on the pan and then transfer them to a wire cooling rack.