To make limoncello: Using a vegetable peeler, remove the peel from the limes in long strips (reserve the limes for another use). Place the lime peels in a 1-quart jar. Pour the vodka over the peels and screw on the lid. Steep the lime peels in the vodka for 1 week at room temperature.
Stir the sugar and water in a large saucepan over medium heat until the sugar dissolves, about 5 minutes. Cool completely, then stir in the vodka mixture. Strain the liquids into bottles, pressing on the peels to extract as much liquid as possible. Seal the bottles and refrigerate until cold, at least 4 hours and up to 1 month.
To make spritzers: Fill 4 tall glasses halfway with ice. Add the mint leaves. Using the handle of a wooden spoon, coarsely crush the mint leaves into the ice. Pour some limoncello over, and stir to blend. Add a splash of club soda and serve.