Preheat the oven to 350 degrees F. Oil an 8-by-8-inch baking pan.
Put the figs and dates in a medium bowl and cover with very hot water until soft, about 5 minutes. Drain and transfer to a food processor; process until very smooth.
Put the chocolate in a small microwave-safe bowl and heat for 30 second increments, stirring after each, until melted and smooth.
Whisk together the flour, cocoa, salt and baking soda in a small bowl.
Add the melted chocolate and oil to the fruit puree and pulse to combine. Add the eggs and vanilla and pulse to combine. Add the flour mixture and pulse until just combined. Then add 1/2 cup of walnuts and pulse until the walnuts are in pieces about the size of peas.
Transfer the brownie batter to the prepared pan and smooth the top. Break up the remaining 1/4 cup walnuts into small pieces and push into the top of the batter. Bake until a toothpick inserted in the center comes out moist with a little bit of batter clinging to it, 25 to 30 minutes. Let cool completely in the pan on a rack. Cut into 16 squares. Store at room temperature in an airtight container for up to 3 days.
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