Combine kiwifruit, lemon juice, and sugar in a small food processor or blender and puree until smooth.

In a cocktail shaker, combine about 1/3 cup of the kiwi puree with 1 oz bourbon and 1 egg white (or 2 tbsp pasteurized whites). Fill shaker with ice and shake, shake, shake until the outside is frosty, then shake a bit more. Taste, and add more lemon or sugar as desired. Pour into cocktail glasses of your choice and serve with an additional half kiwifruit pushed onto the rim.