Special equipment: a 6-cup oval gratin dish, preferably with a wider top than bottom (9-by-6-inches on the bottom, 11-or 12-by-8-inches across the top)
Preheat the oven to 350 degrees F. Generously butter the bottom and sides of the gratin dish. Using a fork, beat the egg and 1/4 cup water in a small bowl until combined.
Put the flour and sugar in the bowl of a food processor and pulse until the mixture is uniform. Drop in the butter cubes, toss with a fork to coat in the flour mixture and pulse until the mixture is crumbly and moist, about 8 long pulses. Add half of the egg mixture and process about 15 seconds. Add the rest and process another 15 seconds. The batter will be homogenous and not too thick. If it curdles a bit, don't worry--that's caused by the brown sugar, and it won't affect the outcome.
Spread the batter in the pan. Press the figs into the batter. Bake until a cake tester inserted in the middle of the cake comes out clean, 30 to 35 minutes. Cool slightly and serve.
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