Preheat oven to 350 F. Line baking sheets with parchment paper or silicone baking mat.
Melt butter in a heavy saucepan or in a microwave-safe bowl.
Add sugars to a large bowl.
Pour melted butter over sugars.
Stir to combine butter and sugars. Let stand 5 minutes.
Add egg and egg yolk; stir to combine.
Stir in vanilla extract.
Add flour, cocoa powder, baking soda, and salt. Stir until just combined.
Stir in chocolate chunks and marshmallow bits.
Chill the dough for at least 30 minutes to give the ingredients a chance to meld.
Using a medium cookie scoop (1.5 tablespoon), drop scoops of dough 2 inches apart on prepared cookie sheet.
Bake for about 8-12 minutes. The cookies should look set at the edges and slightly underdone in the center. Slightly underbaking the cookies will help them to stay soft.
Let cool on cookie sheet for 5 minutes. Remove to wire rack to finish cooling.
Notes
Properly measuring flour makes a huge difference! To properly measure the flour, sift the flour to break it up. Lightly spoon it into the measuring cup and level.
Marshmallow bits are freeze-dried marshmallow pieces. They are usually sold next to the regular marshmallows.