Preheat the oven to 200C/400F/Gas 6.
For the tarte tatin, heat the olive oil in a small ovenproof frying pan, add the tomatoes cut-side down and sprinkle over the sugar. Fry gently for 3-4 minutes until slightly softened.
Cut a circle of puff pastry large enough to fit over the top of the frying pan and place the pastry over the tomatoes. Push the pastry down around the edges to seal in the tomatoes.
Transfer to the oven and bake for 8-10 minutes, or until the pastry is golden-brown. Remove from the oven and allow to cool slightly, then place a plate over the top of the frying pan and carefully turn the tarte tatin out onto the plate.
Heat another small frying pan and fry the goats' cheese on both sides until lightly browned. Place the goats' cheese on top of the tarte tatin.
For the basil oil, place the spinach into a bowl and cover with boiling water to wilt the spinach. Drain the spinach and squeeze out any excess liquid.
Place the spinach, oil and basil into a small food processor. Season, to taste, with salt and freshly ground black pepper and blend until smooth.
To serve, drizzle the basil oil over the tarte tatin.