Melt butter in a large saucepan over medium heat, add leeks, and cook, stirring frequently, until softened, about 10 minutes.
Add 4 cups water, rice, and salt. Cover and cook at an even simmer (lift lid to check and lower heat if necessary) until rice is very soft, about 20 minutes. Stir in lovage and romaine. Bring to a boil over high heat and cook 2 minutes.
Blend soup, in batches, in a blender until very smooth. Return soup to pan, stir in cream, and bring to a gentle simmer. Ladle into bowls, season with salt to taste, and garnish with remaining lovage leaves.
*You can substitute flat-leaf parsley for the lovage, but increase the quantity to 1 1/2 cups (1 oz.).