With a sharp paring knife, cut the figs in half top to bottom, right through the stem, so their natural shape is preserved.
Preheat the oven to 400°F. Scatter a few thyme branches in the bottom of a shallow earthenware dish (or two) just large enough to hold the figs. Place the figs cut side up in the dish. Spoon a few drops of olive oil over them.
Bake the figs for about 20 minutes, until they puff a little and look juicy. Serve the figs on the duck platter, warm or at room temperature.