Scrub oysters under running water; discard any that are open and do not close within a few seconds.
Grill oysters over high heat (400° to 500°) about 5 minutes, until shells open slightly. Pry off top shell, using oven mitts to protect hands from hot shells and liquid. Discard top shell, and loosen meat from bottom shell.
Place on a serving tray, and serve immediately with hot sauce, Champagne Mignonette, and Basil-Garlic Butter Sauce.