1 (2-inch / 5 cm) piece fresh ginger, peeled and sliced in 1?4-inch / .5 cm coins
5 sprigs fresh mint
3 Persian cucumbers, 2 grated (about 1 cup) and 1 thinly sliced
1 (2-liter) bottle chilled soda water
Juice of 1 lemon
In a small saucepan, combine the water, sugar, ginger, and 10 leaves of the mint, torn or chopped into small pieces. Bring the mixture to a boil, and then reduce the heat to medium and simmer for 2 minutes. Remove the syrup from the heat and stir in the grated cucumber. Place the lid on the pot and steep for 30 minutes. Strain the syrup through a fine mesh sieve and set it aside to cool completely.
To make the spritzer in a pitcher, ideally a glass pitcher so the beautiful spritzer is visible, muddle a few mint leaves in a 3-quart / 3 L pitcher. Add the simple syrup, 3 to 4 cups / 700 to 950 mL of ice cubes, a few cucumber slices, the soda water, and the lemon juice. Fill 8 tall glasses with ice cubes, cucumber slices, and mint leaves.
Pour the spritzer into each glass and serve immediately. To make the spritzer in individual glasses, muddle a few mint leaves in the bottom of each glass. Pour a couple of tablespoons of the cucumber syrup into each, and then add ice, a few cucumber slices, and a couple of mint leaves. Pour the soda water into the glasses with a squeeze of lemon juice; stir and serve immediately.