Preheat oven to 350º. Melt chocolate in a small heatproof bowl set over a small pot of gently simmering water over medium-low heat, stirring occasionally with a wooden spoon. Remove bowl from heat and set aside until chocolate is cool. Meanwhile, grease an 8" square cake pan with 1 teaspoons of the butter and set aside. Sift flour and salt together in a small bowl and set aside. Stir baking soda into buttermilk in another small bowl and set aside.
Beat sugar and the remaining 5 tablespoons butter together in a large mixing bowl with an electric mixer on medium speed until light and fluffy, about 2 minutes. Beat in egg yolks, then add melted chocolate and beat until thoroughly combined. Add one-third of the flour mixture, then one-third of the buttermilk mixture, beating well after each addition. Repeat process to use all of both mixtures, then pour batter into prepared pan and bake until a toothpick inserted in center of cake comes out clean, 40–50 minutes. Transfer cake to a rack to cool in pan, then invert onto a cake plate.
For the frosting: Sift confectioners' sugar and cocoa together in a medium bowl. Stir in butter, then coffee, then vanilla, mixing well with a wooden spoon after each addition, until frosting is smooth. Ice top and sides of cake with frosting.