Leaving the fat on, slice each duck breast across the grain into 6 equal pieces. Using a meat mallet, pound the pieces into "scaloppini", 1/8-inch thick and about 4 inches long. Season the flour with salt and pepper, and dredge the scaloppini in the seasoned flour.
In a 10 to 12-inch saute pan, heat the olive oil over high heat until smoking. Add the duck pieces and cook, without turning, until deep golden brown on the first side. Add the cherries, grappa, wine, stock, and butter and bring to a boil. Cook until the liquid is reduced by half, 6 to 7 minutes, then turn the duck pieces over and cook for 30 more seconds.
Transfer the duck to warmed serving plates, sprinkle with scallions, and serve immediately.