Preheat oven to 350 degrees F (175 degrees C). Line a 9x13-inch baking pan with parchment paper, leaving an overhang on 2 sides.

Stir water and flaxseed meal together in a bowl. Let stand until thickened, about 5 minutes.

Mix coconut flour, walnuts, cocoa powder, baking powder, baking soda, and salt together in a bowl.

Mix flaxseed mixture, 2 cups almond milk, apple puree, peanut butter, agave nectar, and vanilla extract together in a large bowl. Add coconut flour mixture; mix until a batter forms. Pour batter into the lined pan.

Bake in the preheated oven until top is dry and edges have started to pull away from the sides of the pan, 40 to 50 minutes. Let cool to room temperature, about 30 minutes.

Drain cashews; add to a blender. Add 2/3 cup almond milk and carob powder; blend until a smooth frosting forms.

Lift cooled brownies out of the pan using the parchment paper overhang. Spread frosting on top. Cut into squares.