Bring a medium pot of water to a boil and season generously with salt. Add the fava beans and cook until tender, about 5 minutes. Using a slotted spoon, transfer the beans to a bowl of ice water to stop the cooking. Peel off the black eyelets from the bean's shells. Squeeze and pop the beans from their shells. Transfer the beans to a bowl.

Add the asparagus and peas to the water and cook until tender, about 4 minutes. Using a slotted spoon, transfer the vegetables to the ice water to stop the cooking and drain. Transfer the vegetables to the bowl of beans.

Preheat the oven to 200 degrees F.

Rub the inside of the skillet with the 1 teaspoon of butter and sprinkle with the shallots. Generously season the fish all over with the salt and pepper, and arrange the fillets, skin-side up, in the skillet. Scatter the mussels over the top and pour in the vermouth and water. Cover the skillet and bring to a boil. Turn the heat to low and cook until the mussels have open (only uncover the skillet once or twice). Remove the skillet from the heat and let rest, covered, until the fish is just cooked through, about 6 to 10 minutes.

Transfer the fish and mussels to a baking pan, leaving the broth in the skillet, and keep warm in the oven. Meanwhile, bring the broth to a boil, add the vegetables and whisk in the 4 tablespoons of butter, little by little, until fully incorporated. Whisk in the herbs.

Divide the fish and mussels among 4 wide, shallow soup plates or bowls. Divide the broth and vegetables among the plates and serve immediately.