Preheat the oven to 375℉.
In a medium saucepan, add the vegan butter, water, soy milk, salt and sugar and heat to medium. Stir with a wooden spoon or spatula until the butter has melted, then bring it a simmer.
Reduce the heat to low and add the flour all at once. Stir until the flour is mixed in and a ball of dough forms. Cook the dough in the pan for 2 minutes, mashing the bottom against the bottom of the pan.
Transfer the dough to the bowl of a stand mixer (or a large bowl) and let cool for 5 minutes.
Prepare egg substitute: In a medium bowl, pour in the 1/4 cup aquafaba, then add the potato starch, tapioca starch, baking powder and xantham gum. Using a whisk, mix the powder into the aquafaba until it becomes foamy, about 1-2 minutes. (SEE PHOTOS in post above for visual reference.)
With the stand mixer on low speed, with the paddle attachment, add nearly all of the egg replacer the the dough mixture, leaving just a tablespoon or two in the bowl for brushing on the pastry before baking. Alternatively, you can use a hand mixer, or even a food processor. Mix until combined, about 30 seconds to 1 minute.
Line a baking sheet with parchment paper and lightly brush it with water, which helps them puff up and not dry out. Transfer the pastry dough to a piping bag with a large tip (Wilton 1A or other large tip). For cream puffs or profiteroles, pipe mounds about 2 inches tall, a few inches apart on the parchment paper. For eclairs, make them mini, about 1 inch tall and 3 inches long.
Brush a little egg replacer on each one, gently. Bake in the center rack of the oven for 20 minutes at 375℉, then without opening the door, lower the heat to 350℉ and bake for 10 more minutes until golden. Remove from oven and transfer to a cooling rack. Allow them to cool completely before slicing open and filling.