Divide the pastry in 2 and roll each piece into a square about 15 x 15 cm (6 x 6 inches). With a knife, lightly mark a border around each pastry square about 1 cm (½ inch) in from the edge and transfer to a baking sheet.
Make the pesto. Put the basil, pine nuts, Parmesan, garlic and oil in a blender or food processor. Season to taste with salt and process until combined but still retaining some texture. (Any leftover pesto can be stored in the refrigerator for a couple of days.)
Mix together the tomatoes, olives, feta and oil, season well and spread over the pastry squares, keeping within the border.
Drizzle 2 teaspoons pesto over the tartlets and bake in a preheated oven, 220°C (425°F), Gas Mark 7, for about 15 minutes until golden and puffed up. Serve with a crisp green salad, if liked.