Line two sheet pans with parchment. Spread 1 cup of the mini marshmallows on a plate and place in the freezer for 10–15 minutes (this helps them hold their shape). Preheat the oven to 360°F (182°C).
In a large bowl, whisk together the flour, cocoa powder, baking soda, baking powder, fine sea salt, and espresso powder until evenly combined.
In a separate bowl, whisk the cooled melted butter with the dark brown sugar and granulated sugar until glossy and thick, about 1 minute. Whisk in the egg, egg yolk, and vanilla until smooth.
Tip the dry mixture into the butter mixture and fold with a spatula just until no dry streaks remain. Stir in the chocolate chips. Cover and chill the dough for 15–20 minutes to firm slightly for shaping.
Scoop dough into 18 portions (about 1½ tablespoons each). Flatten one portion in your palm, place 3–4 of the frozen marshmallows in the center, then wrap and pinch to seal. Set seam-side down on the prepared pans, spacing 2 inches apart. Repeat. Chill the filled dough balls on the pan for 10 minutes while the oven finishes heating.
Bake one pan at a time for 8 minutes. Pull the rack out, quickly press 2–3 of the remaining (unfrozen) mini marshmallows onto the tops of each cookie, and add a few extra chocolate chips if you like. Rotate the pan and bake 2–4 minutes more, until edges look set and the tops are just matte with gooey marshmallow showing.
Cool on the pan for 5 minutes, then move cookies to a rack. While warm, sprinkle lightly with flaky sea salt (optional). Repeat with the second pan.