Preheat oven at 230°C hot-air grill.
Place the giltheads in cold water, clean, scale, dry and place in a roaster.
Cover the fish with rosemary, thyme and garlic, season with salt, pepper, a little grated lemon peel and olive oil.
Peel the green asparagus on the underside, place in a roaster, season with salt, pepper, sugar and olive oil.
Peel and halve the potatoes and cook them in salted water until soft.
Place the giltheads and asparagus in the oven and grill for 25 minutes.
Cut the lovage finely into strips, froth the butter in a pan and brown lightly. Add the lovage, salt, pepper, nutmeg and cayenne pepper to the butter and toss the potatoes.
Arrange the giltheads and asparagus with the potatoes and serve quickly.