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Toss together the flour, 1/2 cup almonds, and sugar. Cut the butter into the dry mixture until it is the consistency of fine bread crumbs. Add 1 egg and a pinch of salt and mix well, kneading slightly. Form the pastry into a ball, wrap in plastic wrap, and refrigerate 30 minutes.
Roll out the dough to 1/8-inch thickness. Line the tart pan with the dough, letting it hang over by 1-inch. Trim the edges, reserving the scraps and rolling them into a ball. Place the pan in the refrigerator for 30 minutes.
Roll the reserved dough ball out to 1/8-inch thickness and, using a pastry cutter with a crimp edge, cut it into several strips that are 9-inches long and 1/4-inch wide. Set these strips aside in a cool place.
Preheat the oven to 375 degrees F.
Beat the egg whites until they are thick and glossy. Spread the apricot jam in an even layer inside the tart crust. Fold the remaining almonds and the chocolate into the whipped egg whites and spread this mixture over the apricot jam. Arrange the dough strips over the top of the tart in a criss-cross fashion, pushing the ends into the apricot mixture. Brush the top with the beaten egg and bake 30 minutes. Serve warm or room temperature.