For the infused rum: Place the chile flakes in the bottle of rum, seal and allow to steep for 24 hours, inverting the bottle every so often to agitate the chiles. Strain out the chile flakes and store the rum in an airtight bottle.
For the mojito: In a cocktail shaker or mixing glass, lightly bruise the mint leaves with a muddler. Add the lime juice and sugar, and stir to dissolve. Add 1 cup of the infused rum and a few ice cubes; stir to combine (about 30 stirs). Top off with club soda and stir 3 times. Divide the cocktail among 4 glasses. Garnish with a mint sprig and serve with a straw.