Brown the butter. In a medium frying pan or saucepan, melt the butter over medium heat, stirring frequently. Once the butter has melted, it will begin to bubble, foam, and eventually brown. (Stir constantly, and be patient! This takes about 6-8 minutes in total.) As soon as you notice a golden brown color and nutty aroma, remove the pan from the heat and pour the browned butter into a large heatproof mixing bowl. Let it cool completely for 30 minutes to 1 hour at room temperature. Do not skip this step! To speed things up, I sometimes stick the bowl in the fridge for 15-20 minutes.
Mise en place. Meanwhile, measure and prepare all ingredients. Preheat the oven to 350 degrees Fahrenheit. Line two baking sheets with parchment paper.
Prepare the dry ingredients. In a medium bowl, whisk together the flour, baking soda, and salt. Set aside.
Mix the wet ingredients. Once the brown butter has cooled completely, add the light brown sugar and granulated sugar. Whisk until combined. Add the eggs and vanilla. Whisk until smooth.
Add the dry ingredients. Add the flour mixture to the wet ingredients, folding with a spatula until a dough forms and the flour is just combined. The dough will be greasy, soft, and a bit sticky. Add the chocolate chips, carefully folding until just combined.
Chill the cookie dough (optional, but recommended). The cookie dough will be very sticky and tough to work with right away. See the recipe notes for more details on the benefits of chilling the dough. Take this time to clean up!Cover the dough with plastic wrap and chill in the refrigerator for 2 hours or up to overnight. To speed things up, I sometimes place the covered bowl in the freezer for 15-30 minutes maximum. Portion and shape the cookies. Using a medium-sized cookie scoop (I used #40 ~2 Tablespoons), scoop balls of dough onto the prepared baking sheets. Place the cookies ~3 inches apart (they will spread quite a bit). I like to do 6 on each sheet to leave room for reshaping after baking.In your hand, flatten each dough ball and add 2 mini marshmallows in the middle. Shape the dough back into a ball, ensuring the mini marshmallows aren't showing on the outside (otherwise they will create melted pools of sugar during baking). If you bake the cookies immediately, lightly wet your hands before this step. This will make dealing with the sticky dough a bit easier.Place the cookie dough balls back on the baking sheet. Add a couple of chocolate chips (I load them up with ~5 per cookie) and mini marshmallows (I like to rip them in half and use 3 halves per cookie) to the top of each before baking, just lightly pushing them in. Bake. Bake one pan at a time for 10-13 minutes or until the edges are just set but the centers are still gooey and look underbaked. Be careful not to overbake! This will keep your cookies super soft.
Reshape and cool. Remove the pan from the oven. Immediately "scoot" the cookies, reshaping them into perfect circles. Check out the recipe notes and video for tips on how to do this! Let the cookies cool on the baking sheet for 10 minutes, as they'll be very soft and gooey before firming up. Transfer the cookies to a wire cooling rack and cool completely.