MAKE THE CAKE: Preheat the oven to 375 degrees. Line a 9-by-13-inch baking pan with parchment paper and coat lightly with nonstick vegetable oil spray. In a medium bowl, stir the pastry flour with the bread flour and baking powder. In a large bowl, mix the egg yolks with the olive oil, water, thyme, salt, and vanilla and 3/4 cup plus 1 tablespoon of the sugar. With a handheld electric mixer, beat the egg yolk mixture at medium speed until very frothy, about 3 minutes. Add the flour mixture and mix at low speed until the flour is fully incorporated.

In a clean bowl, using clean beaters, beat the egg whites at medium-high speed until foamy. Gradually add the remaining 1/4 cup plus 1 tablespoon of sugar and beat until the egg whites are thick and glossy, about 4 minutes. Scoop a cup of the beaten egg whites into the batter and stir until combined. Fold the remaining egg whites into the batter until no streaks remain. Scrape the batter into the prepared pan and bake for 35 minutes, until the cake is golden and starts to pull away from the side. Set the pan on a rack and let the cake cool completely, about 1 hour.

MEANWHILE, PREPARE THE FIGS: In a medium bowl, toss the figs with the sugar, olive oil, chopped thyme, black pepper, and salt. Let stand at room temperature for 1 hour, until the figs begin to soften and release their juices.

Cut the cake into 12 rectangles and transfer to plates. Spoon the figs and their juices over the cake slices, top each slice with a dollop of creme fraiche and a thyme sprig and serve at once.

The cooled cake can be covered tightly with plastic wrap and kept at room temperature for up to 1 day.