Fresh ginger in the mixer and a crystallized ginger garnish make for a sweet and spicy drink.
Mixer: In a blender, puree 1/2 cup warm water, 1/3 cup sugar, and 3 tbsp. coarsely chopped peeled fresh ginger on medium speed for 30 seconds. Scrape down the sides. 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. Add 1 1/4 cups cognac. Refrigerate until cold, about 2 hours or up to 1 week. Up to 2 hours before serving, add 3/4 cup fresh lemon juice and keep refrigerated. Stir well before using.
Cocktail: In a Champagne flute, add 1/4 cup of the mixer. Carefully top with chilled dry sparkling wine (such as Cava). Stir gently. Garnish with crystallized ginger.