Brew tea (make it a bit stronger than usual), then pour into a 2-quart saucepan. Add sugar and stir until dissolved over medium heat to make a simple syrup. Bring to a boil, then turn off heat.
Finely grate the peeled cucumber into the syrup, then steep for 15-20 minutes. Strain out cucumber shreds.
When ready to serve, combine two cups syrup with six cups of club soda. Squeeze in the juice from half a lime, then stir. Pour over ice and float a few cucumber slices in the glass or garnish with a slice of lime.
Makes a 2-quart pitcher, with some syrup left over.