For the ice block:
- 1The day before serving the punch, fill a small saucepan three-quarters of the way full with water. Crush the greens with your hands until wilted and add them to the saucepan.
- 2Cover the pan tightly with plastic wrap, then punch the mint sprigs through the plastic so they’re held upright in the pan. Place in the freezer and freeze until solid.
For the punch:
- 1Combine the gin, Chartreuse, lemon juice, and bitters in a 5-quart plastic or metal bowl. (At this point the mixture can be refrigerated for up to 8 hours.) When ready to serve, nest the punch bowl inside a wider 10-quart plastic or metal bowl (do not use glass or ceramic bowls for either part of this setup, because the cold from the dry ice can crack them). Gently stir in the apple cider and club soda, being careful not to overmix or your punch will be flat.
- 2Remove the saucepan with the ice block from the freezer. Cut away and discard the plastic wrap (do not pull it over the mint sprigs). Dip the outside of the saucepan in hot water or hold it over the flame on a stovetop burner to release the ice block. Remove the ice block from the saucepan and place it in the punch.
- 3Pour the water into the wider outer bowl. Using tongs or gloves, place 1 piece of the dry ice into the water. Do not put the dry ice directly into the punch. Store the remaining dry ice in a well-ventilated area or outside and add additional pieces to the outer bowl as needed for the smoke effect. Serve immediately.