Preheat oven to 350°F. Grease or line a 9x9 square baking pan with parchment paper.
To make the flax eggs, mix together the ground flax seeds and water. Set aside for 5 minutes to thicken.
In a large bowl, whisk together the melted butter and brown sugar. Add the flax eggs and vanilla extract and whisk until combined.
Add the rest of the ingredients into the bowl and stir with a spatula or wooden spoon until combined. You should have a pretty thick batter.
Pour and spread out into the prepared pan. Bake for 20-25 minutes, until toothpick inserted into the center comes out clean. Allow to cool before cutting and serving.
To make the icing, whisk together the powdered sugar and non-dairy milk in a small bowl. Drizzle on top of the blondies.