Preheat the oven to 350°. Butter an 8-inch-square baking pan and line the bottom with wax paper or parchment. In a food processor, finely grind 1 cup of the walnuts with 1/4 cup of the granulated sugar and 1/2 teaspoon of salt.
In a medium bowl, using an electric mixer, beat 4 tablespoons of the butter with the remaining 3/4 cup of granulated sugar until pale and fluffy. Beat in the egg yolks, 1 at a time, making sure each is incorporated before adding the next. Stir in the ground walnuts.
Beat the egg whites with a pinch of salt until stiff peaks form. Gently fold the beaten whites into the walnut batter, then pour into the prepared pan. Tilt the pan to evenly distribute the batter. Bake for about 40 minutes, or until the cake is set and begins to pull away from the side of the pan. Transfer to a rack and let cool for 10 minutes. Run a knife around the side of the pan and turn the cake out onto a serving plate to cool completely.
In a food processor, pulse the remaining 1 cup of walnuts with the confectioners' sugar until finely ground. Add the remaining 4 tablespoons of butter and pulse until creamy. Spread the frosting evenly over the cake. Using the edge of a tablespoon, make a decorative pattern in the frosting. Cut the cake into 1-by-2-inch bars and serve.
An Armagnac's hints of walnut and dark toffee will echo the same flavors in this cake.