In a light-colored saucepan, melt the butter over medium-low heat, swirling the pan occasionally, until it starts to foam. Use a heatproof rubber spatula to scrape the bottom and sides of the pan occasionally to prevent the milk solids in the butter from burning. Cook for 5 minutes, or until the butter smells nutty and is amber with dark flecks at the bottom of the pan. Remove from the heat and immediately pour the brown butter into a liquid measuring cup. Set it on a wire rack to cool while you prep the other ingredients.
Position a rack in the center of the oven and preheat the oven to 350°F. Line two half sheet pans with parchment paper.
Once the butter has melted completely, pour it over the sugar in the mixer bowl and beat on low until just combined. Add the egg and vanilla and beat until just combined. Use a rubber spatula to scrape down the bottom and sides of the bowl. With the mixer on low, gradually add the dry ingredients and beat until just combined. Scrape down the bottom and sides of the bowl once more, and beat on low for an additional 30 seconds. Increase the mixer to medium-low, add the chocolate and graham crackers all at once, and beat until both is evenly distributed throughout, about 1 minute.
Use a 3-tablespoon cookie dough scoop to portion the cookie dough into balls, placing them at least 3 inches apart on the prepared sheet pans. Lightly place a marshmallow directly on top of the center of each cookie dough ball.
Bake one pan at a time for 10 minutes, or until the edges have set but the centers are still gooey. The cookies will look puffed when you pull them out of the oven, but will fall and crack into the perfect cookies as they cool. Cool the cookies on the pan on a wire rack for 20 minutes, or until the edges and bottoms of the cookies have set and feel firm to the touch. Repeat with the remaining cookie dough. Before serving, use a stiff metal spatula and run it underneath the bottom of each cookie to unstick any melted marshmallow. Serve warm or at room temperature. The cookies can be stored in an airtight container or zip-top bag at room temperature for up to 3 days.