Heat the butter and 1 tablespoon of the oil in a shallow sauté pan. Put the fennel in the pan and fry until golden brown on 1 side. Add the vermouth and measurement water, then cover and cook over a low heat or in a preheated oven, 160°C (325°F), Gas Mark 3, for 20 minutes, or until the fennel is tender. You may need to add a little more water if the pan becomes too dry.
Heat a frying pan, add the fennel seeds and chilli and fry for 1 minute until fragrant. Crush them using a pestle and mortar until the chilli has broken up and the fennel seeds are lightly crushed.
Score the skin of the fish and drizzle with a little oil. Sprinkle over the crushed fennel seeds and chilli. Season the fish with salt and pepper. Heat the remaining oil in a hot frying pan and place the fish in the pan, skin-side down. Fry the fish for about 4 minutes, or until the skin is crispy, then turn the fish over and cook for a further minute on the other side.
Place the braised fennel in the centre of the plate and top with the fish. Sprinkle over the reserved herby tops of the fennel and drizzle with a little of the sun-blushed tomatoes, vinegar and any braising juices.