Preheat the oven to 350°F.
Grease the bottom and sides of a 10 x 15 inch half sheet pan (Notes 1 and 2).
Place one cup of the water, the vegan butter, and the cocoa powder in a small saucepan and set it over medium heat. Heat the ingredients up while whisking occasionally, until the butter is melted and the ingredients are fully combined.
Remove the saucepan from heat and set it aside. Let it cool until it's at or close to room temperature (Note 3).
Stir the flaxseeds and remaining ⅓ cup of water together in a large mixing bowl. Set this mixture aside for five minutes, to gel. You can move on to the next step of mixing the dry ingredients in the meantime.
Whisk the flour, baking soda, cinnamon, and salt together in a large mixing bowl.
Add the sugar, yogurt, and vanilla to the bowl with the flax mixture. Beat the ingredients together for about 1 minute using an electric mixer at high speed, until blended and slightly frothy.
Using the mixer at low speed, beat about half of the flour mixture into the flax mixture, followed by about half of the cocoa mixture, followed by the remaining half of the flour mixture, then the remaining half of the cocoa mixture.
Pour the batter into the prepared baking pan.
Place the pan into the oven and bake the cake for 20 to 23 minutes, until a toothpick inserted into the center comes out clean.
Place the cake onto a cooling rack when it comes out of the oven.
When the cake is almost cool, make the frosting. Place the vegan butter, non-dairy milk, and cocoa powder into a small saucepan and place it over medium heat. Heat the mixture up, stirring or whisking constantly, until the butter is completely melted. Remove the pot from heat.
Stir the powdered sugar and vanilla into the mixture, followed by the pecans.
Spread the frosting evenly over the cake. Let it set for at least 10 minutes, then slice and serve.