Simmer champagne (300 ml) and shallot (2) in a small saucepan over medium-high heat for 5-8 minutes until reduced by half.
Reduce heat to medium-low. Add cream (300 ml) and any oyster (18) juice, season with white pepper, and simmer 20 minutes until reduced by half (about 300ml).
Preheat grill to high. Divide oysters (18) in shells among 6 gratin dishes, pour over sauce, and grill 3 minutes until golden and bubbling. Cool 2 minutes, sprinkle with chives and serve.