Preheat the grill. Halve the cherry tomatoes horizontally, placing them on a very shallow large serving dish or plate.
Slice the mozzarella reasonably thinly.
To make the basil oil, place the basil leaves in a small processor. Warm the olive oil and pour over the leaves.
Season with salt and pepper and blend the basil to a purée. Strain through a sieve and set aside.
Drizzle olive oil over the tomatoes, placing a nugget of sea salt on each half. Grind some pepper on the top and sprinkle with sugar. Place under a hot grill.
Once they begin to sizzle, remove from grill and place mozzarella on top of the tomatoes.
Replace dish under the grill, until cheese has melted.
Serve with basil oil and bread.