Prep the mixer the night before so all you have to do in the morning is add Cava and get sipping.
Mixer: In a blender, puree 9 kumquats (halved and seeded), 1/2 cup gin, 1/2 cup sugar, and 2 tbsp. fresh lemon juice for 30 seconds. Scrape down the sides of the blender. Cover and let stand for 10 minutes. Blend for 30 seconds more, then strain into a medium bowl. Transfer to a large jar, bottle, or pitcher. Cover and refrigerate until cold, about 2 hours or up to 1 day. Up to 2 hours before serving, stir in 1/2 cup fresh lemon juice and 1/3 cup fresh mandarin or tangerine juice and keep refrigerated. Stir well before using.
Cocktail: In a Champagne flute, add 3 tbsp. of the mixer. Carefully top with chilled dry sparkling wine (such as Cava). Stir gently. Garnish with whole or sliced kumquats.