1. In a medium saucepan set over medium heat, whisk together the sugar and water. Heat until the sugar is dissolved, then add the kumquats and lemon juice and bring the mixture to a boil. Reduce the heat and simmer until the kumquats are tender and liquid is syrupy, about 15 minutes. Use a spoon to remove any film that rises to the surface, then transfer the liquid and kumquats to a blender. Blend until smooth, then strain the mixture through a fine-mesh strainer, pour into a container and cool to room temperature. Cover and refrigerate until ready to use.

2. Make the cocktail: In a cocktail shaker, combine ½ ounce of the cooled kumquat puree with the gin and Aperol and dry shake. Add the ice and shake again. Double-strain the cocktail, pouring the cocktail through the cocktail strainer over a fine-mesh strainer into a Champagne glass. Top with sparkling wine and serve immediately.