Put half of the figs and half of the raspberries in a medium saucepan. Add the sugars, cornstarch, salt and butter and stir to mix. Bring the mixture to a soft boil over medium heat, stirring constantly. The fruits will release their juices and the liquid will thicken. Transfer the mixture to a bowl and stir in the uncooked fruit. Stir in the lemon and orange juice. Set aside to cool to room temperature.
Cut a little more than half the dough from the disk, keeping the remaining dough wrapped and chilled until ready to use. On a lightly floured work surface, roll the bottom crust into a 12-inch round. Carefully transfer the dough to a 9-inch, fluted tart pan with removable bottom and gently work the dough into the bottom and sides of the pan, leaving the overhang. Spoon the cooled filling into the tart shell; set aside.
On a lightly floured work surface, roll out the remaining dough into a 10-inch circle. Using a fluted pastry wheel, cut the circle into ½-inch wide strips. Plan on using half of the strips for the vertical and half of the strips for the horizontal. Brush the edges of the filled tart shell with the egg wash.
Place a short strip vertically across the left side of the tart. Place a same size strip horizontally across the top of the tart, crossing the first strip. Continue alternating vertical and horizontal strips until the top is covered. Press the strips into the edges of the tart and trim. (Leftover dough can be used to bake cookies in a 350°F oven for 10 minutes.) Brush the lattice top with the egg wash; reserve the wash for another coat. Place the tart on a rimmed baking sheet to catch any drips during baking and chill for at least 30 minutes.
Preheat the oven to 350°F. Remove the tart from the refrigerator and brush once more with the egg wash. Bake for 45-50 minutes or until the pastry is a rich, golden brown and the filling is bubbling. Transfer to a cooling rack to cool to room temperature. The tart is best the day it is made, but leftovers can be stored in the refrigerator for up to 1 day. Bring to room temperature before serving. Enjoy!