In a saucepan over medium heat, melt the butter. Continue cooking for 3–4 minutes, stirring frequently, until golden brown bits form and the butter smells nutty. Remove from heat and cool for 10 minutes.
Melt the dark chocolate in the microwave using short intervals, stirring until smooth. Let cool slightly.
In a large bowl, whisk together the browned butter, brown sugar, and granulated sugar until smooth and glossy.
Add the egg, egg yolk, and vanilla extract. Whisk until fully combined.
Stir in the melted dark chocolate.
In a separate bowl, whisk together the flour, cocoa powder, cornstarch, baking soda, and salt.
Fold the dry ingredients into the wet ingredients just until combined. Do not overmix.
Fold in the semisweet chocolate chips, chopped dark chocolate, white chocolate chips, and mini marshmallows.
Cover the dough and chill for at least 1 hour.
Preheat oven to 350°F. Line baking sheets with parchment paper.
Scoop dough balls using about 3 tablespoons of dough each and place them 2 inches apart on the baking sheet.
Tuck exposed marshmallows into the dough to prevent burning.
Bake for 10–12 minutes until the edges are set and the centers still look slightly soft.
Cool on the baking sheet for 10 minutes before transferring to a wire rack.