Add mint leaves and lime wedges to a shaker or sturdy glass. Muddle gently until the mint is fragrant and the lime releases juice, but do not shred the mint.
Add sugar and muddle briefly again until the sugar begins dissolving into the lime juice.
Pour in white rum and cream of coconut. Fill the shaker halfway with ice.
Shake vigorously for 10–15 seconds until the outside of the shaker feels cold and frosty.
Fill a highball glass with fresh ice and strain the mixture over it. The liquid should look creamy and lightly green-speckled from mint.
Top with club soda and stir gently to combine without losing carbonation.
Garnish with a mint sprig and toasted coconut flakes. Serve immediately while cold and bubbly.