Heat oven to 200C/fan 180C/gas 6. Use a small sharp knife to make a slit along the side of each chicken breast to form a pocket. Mix together the pesto and mascarpone, then carefully spoon a quarter of the mixture into each chicken breast and smooth over the opening to seal.
Brush a little oil, about 1 tsp, all over each chicken breast and season well. Tip the breadcrumbs onto a large plate and season. Place each breast on the plate and press all over with the breadcrumbs. Place in a lightly oiled shallow baking dish along with the tomatoes (kept together on the vine in a couple of bunches). Drizzle over the remaining oil.
Cook in the oven for 20-25 mins until the chicken starts to turn golden and is cooked through. Scatter over the pine nuts and cook for 2 mins more. Sprinkle with basil leaves and serve with new potatoes or crusty bread.