Heat a pan with a splash of oil and brown the chicken breast fillets quickly over a high heat. Season with salt and freshly ground pepper.
Place the fillets in an ovenproof dish or on a baking tray lined with baking paper. Spread pesto over the tops and scatter the cheese on top - grated or sliced mozzarella.
Add the cherry tomatoes alongside, drizzle with a little oil and season with salt and freshly ground pepper.
Put the dish in a preheated oven as soon as you start on the sauce - and bake at 200°C, conventional oven (top and bottom heat), for about 15-20 minutes - depending on the thickness of the fillets. Then let them rest while you finish the sauce, if the timing doesn't quite line up.
Pasta and sauce
Heat a good splash of oil in the same pan you fried the chicken in, and fry the tomatoes, shallot and garlic over a medium heat until the shallot is translucent. Season with salt and freshly ground pepper.
Add the basil leaves and, if you like, a pinch of chilli flakes, and let it simmer for a couple more minutes.
Meanwhile, put the pasta on in a large pot of salted water and cook until al dente. Remember to save some of the pasta water.
Pour the white wine into the pan and let it reduce well while the pasta continues to cook.
Tip the contents of the pan into a blender together with the burrata, and blend until you have a completely smooth sauce.
Pour the sauce back into the pan, add the parmesan and a little dried basil, and stir until the cheese has melted.
Toss the pasta with the sauce and season to taste with salt and freshly ground pepper. Adjust the consistency with a splash of pasta water if needed.
Divide the pasta between 4 deep plates, top each with a chicken fillet and, if you like, a little extra pesto, and arrange the baked tomatoes on top.
Sprinkle with a little grated parmesan and freshly ground pepper before serving - and fresh basil, if you like.