Combine diced figs and cognac in small bowl. Heat in microwave on high power until hot, about 20 seconds; set aside to cool, about 15 minutes.
Adjust oven racks to middle and lower-middle positions and heat oven to 325 degrees. Combine brown sugar, cinnamon, 1 tablespoon granulated sugar in small bowl; set aside.
Spread bread cubes in single layer on 2 rimmed baking sheets. Bake, tossing occasionally, until just dry, about 15 minutes, switching trays from top to bottom racks halfway through. Cool bread cubes about 15 minutes; set aside 2 cups.
Whisk yolks, remaining 3/4 cup sugar, vanilla, and salt together in large bowl. Whisk in cream and milk until combined. Add remaining 8 cups cooled bread cubes and toss to coat. Let stand, occasionally pressing bread cubes into custard, until cubes are thoroughly saturated, about 30 minutes. Stir in cooled fig mixture and transfer to 13 by 9 inch baking dish.
Spread reserved bread cubes evenly over top of soaked bread mixture and gently press into custard. Using pastry brush, dab melted butter over top of unsoaked bread pieces. Sprinkle brown-sugar mixture evenly over top. Place bread pudding on rimmed baking sheet and bake on middle rack until custard has just set, and pressing center of pudding with finger reveals no runny liquid, 45 to 50 minutes. (Instant-read thermometer inserted into center of pudding should read 170 degrees.) Transfer to wire rack and cool until pudding is set and just warm, about 45 minutes. Serve.