Pour cold water into a large mixing bowl and add sugar. Stir until the sugar is totally dissolved. If it's not dissolved all the way, the sugar will stick to the dry ice and you'll end up with a big clump at the bottom of your bowl. Next, add the root beer extract. Taste and add a Tablespoon or two more sugar if desired.
Carefully break the dry ice into small clumps using a kitchen mallet. Add the ice to the mixing bowl using tongs. With a long handled spoon, gently stir the mixture together. It will start to bubble and the fog from the dry ice will begin to creep over the edges of the bowl.
Continue to stir the mixture until most of the ice has dissolved.
Serve in a large bowl or pitcher. Ladle into mugs or cups. If you’d like the dry ice effect to continue, you can add small pieces to the individual serving cups. However, it should be melted before the root beer is consumed.