Preheat the oven to 180°C/gas mark 4. Brush an 18cm round loose-bottomed cake tin with a little of the oil, then line with 1 sheet of filo pastry. Layer the remaining sheets of pastry on top of each other, brushing with oil between each layer.
Put the crème fraîche, milk, grated hard cheese, eggs and most of the thyme in a bowl and mix together. Stir in the crumbled goat's cheese, season with black pepper, then pour the mixture into the tart.
Arrange the halved figs, cut-side up, over the top and scatter over the remaining thyme. Bake in the oven for 45 minutes, until golden and set. Let the tart cool slightly, then slice and serve.