Preheat the oven to 220°C (425°F/Gas 7). Put the eggplant cubes in a single layer on a tray and sprinkle with a little salt. Set aside for 10 minutes, then rinse well. Put the eggplant on a baking tray lined with baking paper, drizzle with half of the olive oil and cook for about 15–20 minutes, or until tender. Remove from the oven and set aside.
Heat the remaining oil in a large saucepan over medium heat. Add the onion, garlic and thyme and cook for 5 minutes, or until softened. Add the chicken and cook for 5 minutes, stirring often. Add the verjuice and stir for about 30 seconds, or until reduced slightly, then pour in enough water to cover the chicken. Bring to the boil, then reduce the heat and simmer for 1 hour, or until the chicken is tender.
Add the mushrooms and baked eggplant to the pan and add the white pepper and salt. Cook for a further 10 minutes, or until the mushrooms have cooked through, then strain off the liquid into another saucepan, reserving the chicken mixture in a large container. Continue to cook the liquid over high heat, or until reduced by half.
To make the béchamel sauce, heat the milk in a saucepan over low heat so it is just slightly warm (not hot). Melt the butter in a separate saucepan and add the flour, stirring until the mixture catches on the edge of the pan. Slowly pour in the warm milk and whisk until smooth and thick. Remove from the heat and whisk into the vegetable mixture until smooth. Add the parmesan and stir to combine. Pour the sauce over the chicken mixture and stir through, then adjust the seasoning. Allow to cool before stirring through the torn basil leaves.
Roll out the savoury pastry following the instructions and use it to line the base and sides of six 12.5 cm (4¾ inch) pie tins. Roll out the puff pastry following the instructions and cut out six circles with a 13.5 cm (5¼ inch) diameter to make the pie lids.
Spoon the mixture into the pastry-lined pie tins, filling them to the brim. To attach the puff pastry lids, brush the rim of the pastry base and lid with a little egg wash and lay the lid over the base. Pinch gently between your thumb and index finger to make a good seal around the circumference edge. Brush the top of the pie lid with egg wash, sprinkle with sesame seeds and pepitas, and make a small hole in the middle to allow steam to escape. Reduce the oven temperature to 180°C (350°F/Gas 4) and bake the pies for 30–35 minutes, or until golden brown on top. Remove the pies from the tins and leave to cool for a few minutes before serving.