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To make the gratin, peel and slice the potatoes no thicker than a 50p piece using a mandolin, if you have one. Bring the milk, cream, garlic, herbs and some seasoning to the boil in a large saucepan, then turn down the heat and simmer for a few mins.
Slip the potatoes into the hot milk mixture and simmer for 10 mins until just cooked. Drain the potatoes in a colander over a bowl to catch the liquid – reserve the milk. While the potatoes are simmering, heat half the butter in a frying pan until foaming. Fry the mushrooms for 2 mins until just wilted, season with salt and pepper and set aside.
Rub 2 small gratin dishes (or 1 medium) with the remaining butter, sprinkle over the thyme leaves and grate over some of the cheese. Fill the dishes halfway with potato slices, moisten with a little milk and grate over more cheese. Fill up the dishes with potato slices, add enough milk to cover, then top with the mushrooms and the rest of the cheese.
For the chicken, mix about two-thirds of the butter with the lemon zest and half the juice, the parsley, paprika and some salt and pepper. Lift the skin slightly away from each breast, spread or pipe the flavoured butter under the skin, then stretch the skin back over. Chill up to a day in advance, if you like.
Heat oven to 220C/200C fan/gas 7. Heat the remaining butter in an ovenproof frying pan until foaming. Pan-fry the chicken, basting constantly with the butter until starting to brown. Add the shallot, garlic, thyme and stock. Place the chicken, skin-side up in its pan, on the higher shelf of the oven, and the gratins on the lower shelf. Roast both for 25-30 mins, then remove the chicken and rest for 15 mins, while the gratins continue to cook.
To finish, remove the chicken from the pan. Place the pan back on the heat, squeeze over the remaining lemon juice and bring everything to a hard boil. Pass the sauce through a sieve, pressing down firmly on all the soft shallot and garlic. Heat a drizzle of olive oil in another frying pan and reheat the leeks until they start to colour. With the gratins cooked, you are now ready to plate up.
Line six leeks up in a row in the middle of each plate. Carve the chicken on a slant into five thick slices. Neatly fan out each chicken breast over the leeks. Spoon the sauce over the chicken and around the leeks, and serve with the potato gratin.