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To make the tarragon oil, chop the tarragon leaves, then warm the groundnut oil in a small pan. Add the tarragon, season with a pinch of salt, then liquidise until smooth with a stick blender. Set aside. Can be done up to 2 days ahead.
Cook the potatoes in boiling salted water for 20-25 mins. Cooking them a bit longer than usual will give them a very soft, almost creamy texture. Drain and halve the potatoes lengthways and season with salt and pepper.
While the potatoes cook, arrange the tomatoes on a baking tray, season, then drizzle liberally with walnut oil. Put under a pre-heated grill and cook for 6-8 mins until they start to colour and soften. Set aside to be served at room temperature. Add the sliced red onions to the cooked potatoes, with the lime juice. Cover and keep warm. Heat oven to 220c/fan 200c/gas 7.
Rinse the mushrooms briefly and pat dry. Sauté in butter over a high heat, then season. Add the mushrooms to the potatoes and red onions. Stir the mustard into the cream, then lightly whip the cream (see tip, opposite). Season and stir it amongst the potatoes, onions and mushrooms, then spoon over a large ovenproof platter. Lay the strips of cheese, overlapping haphazardly, on top. Can be done up to 1 hr ahead.
Now finish it in style. Bake until the cheese begins to melt and is tinged to a light golden brown, about 10-12 mins. Arrange the grilled tomatoes down either side of the platter, scatter the broken walnuts (if using) Over the tomatoes and drizzle with tarragon oil. Serve the rest of the tarragon oil separately.