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.
Food Network Kitchens suggest caution in consuming raw and lightly-cooked eggs due to the risk of Salmonella or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly-refrigerated, clean, grade A or AA eggs with intact shells, and avoid contact between the yolks or whites and the shell. For recipes that call for eggs that are raw or undercooked when the dish is served use shell eggs that have been treated to destroy Salmonella, by pasteurization or another approved method.