Prepare the flax egg by whisking ground flaxseed meal and water in a small bowl and allowing it to hydrate for 10 minutes until gelatinous.
In a stand mixer equipped with a paddle attachment, cream the softened vegan butter and brown sugar on medium speed until aerated.
Incorporate the hydrated flax egg and molasses into the butter mixture, mixing until fully emulsified.
In a separate bowl, whisk together the flour, baking soda, ginger, cinnamon, cloves, and salt to ensure even leavening distribution.
Reduce mixer speed to low and gradually add the dry ingredients to the wet mixture until a stiff dough forms.
Divide the dough into two flat discs, wrap in plastic film, and refrigerate for at least 60 minutes to hydrate the flour and firm the fats.
Preheat oven to 350°F (175°C) and line heavy-gauge baking sheets with parchment paper.
On a lightly floured surface, roll the chilled dough to a uniform thickness of 1/4 inch.
Cut shapes using cookie cutters and transfer to prepared baking sheets, spacing them 1 inch apart.
Bake for 8 to 10 minutes until the edges are firm to the touch; the centers will remain slightly soft but will firm upon cooling.
Transfer to a wire cooling rack to allow for air circulation and complete setting of the internal structure.