Trim off all but 1-inch of the scallion greens, chop the greens, and reserve for garnish. Cut each of the scallions into 4 pieces. Combine the stock, scallion pieces, ginger, and garlic in a large pot. Bring to a boil over high heat. Reduce the heat to medium and simmer for 20 minutes or longer. Strain and discard the solids. (The broth can be made a day ahead. Keep refrigerated and ready to serve.)
Place the chicken in a food processor and pulse several times to grind finely. Add the ice and process until the ice is absorbed. With the blade in motion, add the cream. Pulse in the lemon zest and salt to blend. Chill for 1 hour. Sprinkle a sheet pan with cornstarch to hold the wontons before cooking.
Working with a few wontons at a time and keeping the remaining wonton skins covered, mound 1 tablespoon of the chicken mixture in the center of each skin. Moisten the edges with water and place another wonton on top. Press on the edges to seal completely. Repeat until the entire filling is used. As they are done, transfer to the sheet pan.
Just before serving, bring a large pot of salted water to a boil over high heat. Working in 2 batches, drop the ravioli into the boiling water, reduce the heat to medium, and simmer until tender and the chicken mixture is thoroughly cooked, 3 to 5 minutes. Bring the water back to a boil before proceeding with the final batch. Remove the wontons with a slotted spoon and arrange 3 in the center of each serving bowl. (The wontons are best if used just after boiling. Place them right into the serving bowls. If they cool, the hot broth will warm them.)
Reheat the broth almost to boiling, season with salt and pepper, to taste, and ladle over the wontons. Garnish with the cilantro and chopped scallion greens. Serve right away.