In a cocktail shaker, combine rum, banana liqueur, lemon juice, thyme syrup and egg white. Shake gently to mix. Add ice to fill, then shake vigorously. Strain into a coupe, sprinkle bitters on top of the foam and garnish with fresh thyme.
For the thyme syrup:
Combine sugar, water and thyme sprigs in a small saucepan and bring to a simmer over medium-high heat, stirring often to dissolve sugar. Once the mixture is simmering, strain into a heat-proof airtight container and refrigerate until ready to use.