Horizontally slice both chicken breasts right through the centre to create 4 even-sized cutlets. Finely dice around 3/4 of the basil, saving a handful of smaller leaves to serve.
In a large shallow dish, combine the flour, parmesan, salt and pepper. One by one, coat the cutlets in the flour mixture, give them a light shake, then place to one side.
In a large pan over medium-high heat, add 1 tbsp butter and oil. Once the butter has melted and the pan is hot, add the cutlets and fry both sides for a few minutes until golden and cooked through the centre, then place on a plate to one side.
Lower the pan heat to medium, then melt in the remaining 2 tbsp of butter. Add the garlic and fry for 30 seconds or so (be careful it doesn't burn), then pour in the stock. Stir in the tomatoes alongside a pinch of salt, and simmer for around 2 minutes until you notice the tomatoes just starting to soften. Stir in the balsamic vinegar and diced basil and simmer for a further minute or so, giving everything a good stir until the sauce binds and thickens a little.
Stir in the resting juices from the chicken, then check for seasoning and adjust with salt and black pepper. Add the chicken and baste in the tomatoes until fully coated in the sauce (it will now thicken a little more due to the floured coating on the chicken).
Serve with extra basil and enjoy!