Bring 1 cup water to a boil in a small saucepan over medium-high heat. Turn off the heat, add the tea bags and let steep 5 minutes. Discard the tea bags. Stir in 1 cup ice cubes until completely melted. Pour the tea into a liquid measuring cup, and let it chill completely in the refrigerator, about 30 minutes.
Meanwhile, cook the sugar and 1/2 cup water in a small saucepan over medium heat, stirring occasionally, until the sugar has completely melted, about 5 minutes. Turn off the heat, add the mint and stir until it has completely wilted. Let cool completely at room temperature, at least 30 minutes.
Strain the cooled mint syrup into a large liquid measuring cup or a pitcher, then press the leaves with a wooden spoon or spatula to extract as much liquid as possible. Discard the mint. Add the blueberries, and use a wooden spoon or a potato masher to lightly crush most of them (a few whole blueberries is fine). Add the chilled tea, and stir to combine.
To serve, fill 4 tall glasses with ice. Divide the blueberry mixture among them. Add enough sparkling wine to fill each glass. Garnish with a mint sprig, and serve immediately.
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