Preheat the oven to 350 degrees. Butter and flour a 9-by-13-inch baking pan. Grate the zucchini on the coarse holes of a box grater. You should have about 2 1/2 cups of grated zucchini. 

On a piece of parchment, sift together the flour, cocoa, baking soda, and salt. In a mixer fitted with the paddle attachment, beat the butter, sugar, and oil at medium-high speed until smooth and light, about 2 minutes. Add the eggs one at a time, beating well after each addition. Beat in the vanilla. Add the flour mixture in several additions, alternating with the buttermilk, beginning and ending with the flour. Beat the batter at high speed about 10 seconds, just to mix thoroughly.

Remove the bowl from the mixer, and stir in the zucchini and walnuts. Pour the batter into the prepared pan. Bake until a tester inserted in the center comes out clean, about 40 minutes. Let it cool completely on a rack before cutting it into squares to serve.