For the cake: Preheat the oven to 350 degrees F and butter an 8-by-8-inch baking dish.
Set aside 1/4 cup each of the apricots and raisins and dice; reserve for later. Put the remaining apricots and raisins in a medium bowl and cover with very hot water, then let soak until the fruit is very soft, about 5 minutes. Drain and puree in a food processor along with the orange juice until very smooth.
Whisk the flour, baking powder, cinnamon and salt together in a medium bowl.
Beat the butter in a large bowl with an electric mixer on medium-high speed until smooth. Add the apricot-raisin puree and beat until incorporated and smooth. Add the egg and egg yolk, vanilla and orange zest and beat until incorporated. Lower the mixer speed to medium-low and add half the flour mixture, then all of the sour cream and then the remaining flour mixture, stopping to scrape down the sides of the bowl as needed (it's ok if the batter has some lumps). Fold in the diced apricots and raisins.
Transfer the batter to the prepared baking dish. Bake until golden, the cake bounces back when pressed in the center and a toothpick inserted in the center comes out clean, 25 to 30 minutes. Cool in the pan on a rack.
For the frosting: Soak the chopped dates in very hot water until very soft, about 5 minutes. Drain and transfer to the food processor. Put the chocolate in a small microwave-safe bowl and heat for 30 second increments, stirring after each, until melted and smooth. Add to the dates along with the heavy cream and cocoa powder. Process until light and smooth.
Once the cake is completely cooled, spread the chocolate-date frosting over the top. Cut into squares and serve.
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