1. Arrange onions in even layer in 10-inch skillet. Stir in sugar and next 4 ingredients (through 2 cups Champagne); bring to boil over high heat. Let mixture reduce 7–8 minutes or until almost all liquid has evaporated.
2. Reduce heat to medium-low; continue cooking, stirring frequently for 7–8 minutes or until onions are tender and golden.
3. Add remaining 2 tablespoons Champagne to pan; scrape browned bits at bottom with spatula. Remove thyme sprig, and transfer onions to serving bowl. Garnish with additional thyme, if desired.