Preheat the oven to 425°F.
Place the walnut pieces in a food processor and pulse the walnuts until they are broken down into crumbs. Add the flour, powdered sugar, white sugar, and salt to the processor bowl. Pulse a few times to combine.
Cut the butter into 1/2-inch chunks and add them to the processor. Pulse again, until the butter is broken down.
The whole mixture should be fairly uniform with tiny pebble sized pieces of walnuts and butter.
Separate the egg and set aside the egg white in a small bowl. Place the egg yolk in a small bowl and beat in 4 tablespoons of water. Drizzle the liquid into the bowl.
Pulse until a dough forms. If the dough seems too crumbly and dry, add 1 tablespoon of water at a time, until a dough forms.
Carefully remove the blade from the processor. Grab chunks of the dough from the processor and press them into the bottom and sides of a 10-inch tart pan with a removable bottom.
Continue to do this with all the dough, until the pan is evenly lined with the dough.
Place a piece of parchment paper over the top of the dough and fill with pie weights or dried beans. Place the pan on a rimmed baking sheet or pizza pan and bake in the oven for 12 minutes. While the crust is baking, beat the egg white with 1 tablespoon of water.
Remove the pie weights by carefully lifting the parchment paper up and moving it with the weights inside to a heatproof bowl. Brush the egg white wash over the sides and bottom of the tart crust. This will help seal the crust and prevent it from becoming soggy. Move the pan back to the oven and bake an additional 5 to 7 minutes, or until the edges of the pan start to brown and the top of the dough looks dry.
While the dough is baking, make the filling by combining the goat cheese with the honey. Cut the figs in half lengthwise.
When the tart dough is done baking, spread the honey goat cheese over the still warm crust.
Then place the fig halves around the edge of the tart, overlapping the fig stems over the body of the figs, in a shingle pattern. Sprinkle the blackberries over the center area of the tart. Place back in the oven for an additional 20 minutes.
Let the tart cool on a wire rack for 15 minutes, then drizzle an additional 1 to 2 tablespoons of honey over the tart and serve.