Lay the chicken breasts in a frying pan with the bay leaves and cover with water. Bring to a gentle simmer, then simmer for 20 minutes until cooked. When chicken is thoroughly cooked with no pink showing, juices will run clear when the meat is pierced or sliced.
Remove and chop the chicken into small cubes. Cook the spaghetti in a saucepan of boiling water according to packet instructions, adding the peas for the last 3 minutes of cooking time. While the spaghetti cooks, heat the oil in a large frying pan, then add the shallots and fry for 3-4 minutes until they are starting to brown. Stir in the garlic and fry for 1 minutes, then tip in the courgette and stir-fry for 2-3 minutes. Stir the chicken into the pan, season with salt and pepper. Keep warm.
Drain the spaghetti in a colander over a bowl, so you can reserve a little of the pasta water, then drop the spaghetti and peas into the pan with the chicken. Mix the eggs and cheese together with some salt and pepper, remove the pan from the heat and pour in the egg mixture. Stir everything together to coat well, spooning in enough of the pasta water to give a sauce. Toss in most of the parsley, then serve immediately sprinkled with the rest of the parsley.
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