Cut a large honeydew melon (about 4 1/2 pounds) into chunks and finely chop enough of the chunks to equal 1/2 cup. In a large bowl, crush 3 large sprigs mint. Using a food processor and working in batches, puree the honeydew chunks, 2 tablespoons sugar, the grated peel and juice of 1 lime, 1/8 teaspoon ginger and a pinch of cardamom; add to the mint. Refrigerate until well chilled. Meanwhile, in a small bowl, combine 3/4 cup greek-style yogurt with the grated peel of 1 orange and 1/4 teaspoon ground cinnamon. Remove the sprigs of mint from the soup. Serve the soup topped with the finely chopped melon, the orange yogurt and fresh mint leaves.