Brush 1 side of each bread slice lightly with butter, then toast on a baking sheet in middle of oven until golden, about 6 minutes, and cool.
Simmer rhubarb, ginger, water, and sugar in a small heavy saucepan, uncovered, stirring gently, until rhubarb is just tender, 2 to 3 minutes, then set pan in a bowl of ice and cold water to stop cooking. Drain rhubarb in a sieve set over a bowl, reserving rhubarb, then whisk together 2 teaspoons rhubarb liquid and mayonnaise. Discard remaining cooking liquid.
Remove skin and fat from duck breast. Carve 18 (1/8-inch-thick) slices from duck, cutting across the grain. Trim slices to fit toasts, reserving any remaining duck for another use.
Spread about 1 teaspoon rhubarb mayonnaise on buttered side of each toast and top with 1 slice of duck, 1 teaspoon rhubarb, and a cilantro sprig.
* Available at D'Artagnan (800-327-8246).
Cooks' notes: · Toasts can be made 1 day ahead and kept in an airtight container at room temperature. · Gingered rhubarb can be made 1 day ahead and chilled in cooking liquid, covered.