I recommend weighing the flour, but if you don't have a scale you can just carefully spoon the flour into a measuring cup and level off the excess. Add the flour, sugar, salt and yeast to a bowl of a stand mixer. Whisk to incorporate.
Make a well and then pour in the lukewarm water (it should be about 110 F degrees). Start off with 3/4 cup and use more as needed.
Add the aquafaba and then gently fold everything together with a spatula.
Add the bowl to a stand mixer with the bread hook attachment. Mix on Speed 2 for 1-2 minutes and then add in the vegan butter.
Let the machine knead the dough for about 6-8 minutes or until a firm but smooth, soft and supple dough ball is formed. It shouldn't be sticky and should hold it's shape.
Add the dough ball to a greased bowl and cover with greased saran wrap or a clean towel. Let it proof and rise for about 1 hour or until doubled in size. I usually do mine in the oven with the light turned on.
Once doubled, punch it down and degas it. Then dump it out onto a floured work surface. Flatten it more and degas it. Then cut it into 8 pieces. They should be about 90 - 100 grams each.
To form the pieces into balls, tuck each corner under itself and pull the edges to form it into a ball. Then rotate the ball in between your thumb and finger. Keep it in constant contact with the cutting board / work surface. Keep rotating the ball to create tension and to close the seam up on the bottom.
Add it to a parchment lined baking sheet and flatten the ball out with the palm of your hand. Repeat with the rest of the dough and then cover them with another sheet tray or clean towel to rest for 10 minutes
After 10 minutes, take each ball and flatten it again, then shape it into a hot dog bun by grabbing it from the top and rolling it over itself. Almost like rolling a burrito. Roll it over about 3 - 4 times, and then gently roll it with your finger and stretch it out to about 6 inches.
Add each roll to a parchment or silicon lined baking sheet. Space them about 1/2 - 1 inch apart. They will rise into each other, but that's what we want. Cover them with another baking sheet or clean towel for one hour for the second rise.
After they have puffed up, brush them with the vegan egg wash of your choice. Then bake in an oven preheated to 375 F for 15 - 18 minutes or until golden brown. Remove from the oven, and brush with some melted vegan butter if you wish. Let them cool on a wire rack for 10 minutes and then dig in!