Heat grill or grill pan. Prepare an ice bath; set aside. Make the sabayon: Combine yolks, sugar, champagne, and liqueur, if using, in a large metal bowl set over a large pan of simmering water. Whisk until mixture is very thick and has expanded in volume, about 7 minutes. Place bowl in ice bath; let cool completely.
Place cream in a large bowl, and beat until stiff peaks form. Fold whipped cream into egg-yolk mixture. Cover with plastic wrap, and place in refrigerator at least 20 minutes.
Line grill or pan with heavy-duty foil. Brush peaches with butter; sprinkle with brown sugar. Grill cut side down until peaches are tender and sugar is caramelized, 6 to 7 minutes. Divide sabayon among four dishes, and top each with a peach half. Serve.