Preheat the oven to 350°F (175°C). Grease a 9×13 inch baking pan and dust it lightly with flour, or line it with parchment paper.
In a large bowl, sift together the flour, cocoa powder, baking soda, salt, cinnamon, nutmeg, and allspice. Stir to combine evenly.
In a separate bowl, beat the eggs with the sugar and vegetable oil until smooth. Gently fold in most of the diced apples and the chopped nuts, saving about 1 cup of apples for the top.
Add the dry ingredients to the wet mixture in small portions, folding gently until just mixed. Take care not to over mix to maintain a tender texture.
Pour the batter into the prepared pan and spread it evenly. Scatter the remaining diced apples over the surface. Bake for 50–55 minutes, or until a toothpick inserted in the center comes out clean.
Allow the cake to cool in the pan for 10–15 minutes before moving it to a wire rack. Slice and enjoy warm, or let it cool fully. Dust with powdered sugar if desired.