Preheat the oven to Gas Mark 6, 200°C, fan 180°C. Mix the soft cheese with the garlic, parsley and lemon zest. Season with pepper.
Cut a pocket into the chicken breasts and put a spoonful of the cheese filling into each. Place on a lightly oiled baking tray. Drizzle all the chicken with 1 tbsp olive oil, scatter over the thyme leaves and season. Bake for 20-25 minutes, until cooked through.
While the chicken is cooking, half fill a large saucepan with water and bring to the boil. Add the spaghetti and cook for 10 minutes, then add the broccoli florets and cook for a further 2 minutes.
Drain the spaghetti and broccoli and return to the pan. Add the remaining olive oil, lemon juice, parmesan, chilli flakes and plenty of pepper. Toss together.
Take the chicken breasts out of the oven. Cut into a thick portion and ensure that it is cooked through, with no pink showing. Divide the spaghetti and broccoli between four plates, top each with a stuffed chicken breast and serve immediately.
For top tips on protecting you and your family when preparing raw meat and poultry, visit Food Safety in the Home and view the full Food safety at home video.