For the rum: Place the rum in a non-reactive container. Wrap the gingerbread in cheesecloth and add to the rum. Allow to infuse overnight. Strain the rum through a cone filter.
For the daiquiri: Combine 2 ounces infused rum, lime juice, cane syrup and egg white in a cocktail shaker. Shake vigorously without ice. Add ice to the shaker and shake a second time. Moisten the rim of a cocktail glass with the lime wedge. Rim the glass with equal parts cinnamon and sugar. Strain the cocktail into the glass.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.