There is a ton of lore that surrounds this cocktail… How much of it is true, is anyone’s guess. What does appear true is that this cocktail was originally made with Bourbon, before the Gin version became more famous.
So here we’ll stick to the original:

Ingredients:
Half a lime
1½ oz Bourbon
Club soda
Glass: Highball, or collins

Method:
Squeeze half a lime into a highball; then drop in the lime shell.
Fill the glass with ice, add the bourbon, and top with club soda.
Stir and serve.