Gently muddle the mint leaves with the demerara simple syrup in a chilled highball or collins glass.
Add the demerara rum and lime juice to the glass. Fill the glass about two-thirds full with crushed ice.
Insert a swizzle stick (or a long barspoon) into the drink, ensuring it reaches the bottom of the glass. Swizzle by rotating the stick rapidly between the palms of your hands. Continue until the outside of the glass becomes well-frosted.
Top the glass with more crushed ice, forming a slight dome on top. Dash the Angostura and Peychaud's bitters directly onto the top of the ice and garnish with a large, fresh mint sprig.