To make 16 cups of white miso syrup: Add 8 cups water and 4 tbsp. white miso paste to a medium-sized pot and heat until the paste is dissolved, about 5-10 minutes. Add 8 cups pure cane sugar and bring to a boil, then turn off heat, stirring occasionally. Let sit until it reaches room temperature.
To assemble the cocktail: Rip the mint into small pieces and place in a cocktail shaker. Add the remaining ingredients, except the soda, and shake.
Pour everything into a Collins glass and top with the soda.
Garnish with mint sprigs and a lime wheel.