Preheat the oven to 180C/160C fan/gas 4.
Heat the oil over a high heat in an ovenproof casserole dish large enough to hold the chicken. Season the chicken with salt and brown it, breast-side down (around 10 minutes, but check colouring after 5). Remove, then add the onions and cook them over a high heat until they are taking on some colour (approx. 3 minutes).
Place the chicken back in the dish, breast-side up. Put a few garlic cloves, a sprig of rosemary and a piece of lemon zest in the cavity of the chicken and add the rest to the pot. Pour in the white wine or vermouth and top with the tomatoes and olives.
Put the lid on the dish and place it in the oven. Cook for 1 hour to 1 hour 30 minutes (depending on the size of the chicken), then remove the lid and cook for a further 10 minutes, until the skin of the chicken has crisped up. Check to see if it the chicken is done – if you have a Thermapen or thermometer, the internal temperature of the chicken should be at least 74C. The juices should run clear, and the legs should feel loose.
Cook the tagliatelle according to packet instructions. Remove the chicken from the dish and place it on a serving dish, with the tomatoes, olives, and onions around it. Squish the flesh out of the garlic cloves and whisk it into the gravy in the casserole dish. Add the tagliatelle to the dish and toss it in the gravy.
Divide the tagliatelle between 4 large pasta bowls and garnish with a few basil leaves. Serve the chicken at the table.