Beat the cream to soft peaks in a chilled bowl; then fold in the horseradish, chives, lemon zest, lemon juice, salt, and pepper.
Taste for seasoning, adding horseradish and lemon juice as necessary; the mixture should be highly seasoned.
If you have a fish or vegetable grate, preheat it on the grill grate (medium-high) for 5 minutes, then arrange the oysters on it, rounded side down—otherwise, arrange the oysters directly on the grill grate; grill 6-10 minutes, until the shells open.
Transfer the oysters to serving plates or a platter, discarding any top shells or oysters that did not open.
Place a spoonful of horseradish cream on each oyster and serve immediately.