Cut tomatoes into eighths, and set aside.
Melt butter in medium-size saucepan over medium-low heat. Add onion; cook 1 minute. Add garlic and sprinkle with allspice; cook 4 minutes longer. Stir in the hot stock, amaranth, tomatoes and sugar. Heat to boiling. Reduce the heat and simmer, partially covered, 25 minutes. Set aside to cool.
Transfer the cooled soup, a third at a time, to a blender container. Blend until very smooth. Pour through a fine-mesh strainer into a bowl. Beat in the ½ cup sour cream, and season with salt and pepper. Refrigerate until cold, 2 hours.
Just before serving, combine the mint and the cucumbers. Spoon some over each bowl as you serve the soup, and dapple with sour cream. Serves 4 to 6.