Place the yeast in a large bowl or the bowl of a stand mixer. Add the nondairy milk and mix. Set aside for five minutes to allow the yeast to bloom.
Add 1 cup flour, aquafaba, sugar and salt to the bowl. Mix with a spatula or using the paddle attachment on the stand mixer.
Gradually add the remaining flour to the bowl and mix by hand or in the stand mixer until all the ingredients have come together. Cover the bowl and let the dough rest 20 minutes.
Once the dough has rested, knead it. You can do this on medium-low speed in the stand mixer with the dough hook attached, for about 10 minutes. Since this is a sticky dough, this is best done in a mixer, but if you do it by hand be patient. It will take about 15 minutes before you get the right consistency. Fold the dough over itself and use a bench scraper to scrape up any that is left behind and add it back to the dough. By the end the dough should look smooth and supple. It will be soft but no longer sticky and it will be quite elastic.
At this point begin adding the butter to the dough. Again, a stand mixer is better for this, but if you're doing this by hand make sure you knead vigorously as you add the butter in. Set the stand mixer to medium-high or high speed and add the butter one tablespoon at a time, making sure you drop the butter on the dough and not the side of the bowl (so it doesn't just flap around. I kept the mixer on low speed to capture the video, but make sure it's at an eight or so on a Kitchen Aid and you'll get the job done much faster.)
Once you've added all of the butter to the dough the dough will look loose and stick to the bottom of the bowl.
Continue to knead until a very smooth, elastic dough forms. It will be soft and shiny but not at all sticky.
Form the dough into a ball and place in an oiled bowl. Let it stand two hours or until doubled.
Once it has doubled, knead it briefly, then shape into a ball again and place in an oiled bowl. Cover tightly and refrigerate eight hours or overnight. This really helps the flavor of the brioche develop.
The next morning remove the dough from the fridge. Shape it into 10 smooth rounds. If making pull apart rolls, as I did, place the rounds in an oiled two-quart baking dish, apart but close to each other. For more rounded buns, place them at least two inches apart on an oiled baking sheet.
Cover loosely with oiled cling wrap and let the buns rise until they are nearly doubled but not quite. This should take about an hour to an hour and a half in a warm kitchen or in summer. My kitchen was quite cool on a fall day and it took nearly two hours.
Preheat the oven to 375 degrees Fahrenheit.
You can brush on an optional glaze of 1 teaspoon maple syrup, 1 teaspoon nondairy milk and 1 teaspoon oil on the tops of the risen rolls for a shiny look and darker color.
Bake the rolls for 20-22 minutes or until the tops are lightly brown. Remember that for more color you will need to add more sugar to the rolls.
Let the rolls cool on a rack. Serve warm or cold.