Put the green tea bags into a heatproof container and pour in 1 quart boiling water. Let steep until it comes to room temperature, about 1 hour. Remove the tea bags and discard.
While the tea is steeping, in a saucepan over medium-high heat, bring 1 cup water, the sugar, and ginger to a boil. Stir to dissolve the sugar, remove from the heat, and allow to cool to room temperature, about 1 hour. Store the tea and ginger simple syrup in the refrigerator until ready to serve.
To serve, fill glasses with ice and pour in the tea. Sweeten, to taste, with the ginger simple syrup. Garnish with a slice of ginger. For the adults, top each with 1/2 cup prosecco, if desired.