Preheat the oven to 220°C/fan 200°C/gas 7 and line a baking tray with baking parchment.
Scatter some flour over a clean work surface and roll out the puff pastry to about the thickness of a £1 coin. Take a side plate roughly 19cm in diameter and lay it upturned on the pastry. Pull a knife all the way around the outside. Repeat. Separate out the discs from the spare pastry and transfer them to the lined baking tray. Reserve leftovers for whatever you please.
Pop the baking tray into the oven and cook the tarts until you estimate them to be half risen. At this point quickly slice the tomatoes.
Remove the pastry discs from the oven and press them down with a spatula.
Turn the oven temperature up to 240°C/fan 220°C/gas 9.
Spread the pesto over the pastry discs in an even layer, leaving a border of roughly 1.5cm of un-pestoed pastry all the way around.
Lightly pat the slices of tomato with kitchen paper to absorb some of the moisture. Working from the centre of each pastry disc, quickly lay the tomatoes on the pastry, overlapping in concentric circles, until totally covering the pesto area – you will need around 15 slices per pastry disc.
Brush the tomatoes with 1 teaspoon of the olive oil. Do not season. Put the tarts back into the oven for a further 8 minutes.
In the meantime, stack the basil leaves and slice them finely. Put them into a small bowl with the 3 tablespoons of olive oil, the garlic, lemon juice and zest.
Remove the tartlets from the oven. The tomatoes should be wrinkly at the edges and not be remotely watery. The uncovered rim of pastry should be golden brown.
Season with plenty of sea salt and pepper and fleck all over with the basil dressing. Serve immediately.