Place the yeast and 1/3rd cup of warm, nondairy milk in the bowl of a stand mixer and let the yeast bloom, about five minutes.
Add the all-purpose flour, aquafaba, sugar, and salt to the yeast mixture. Mix well on medium speed until everything is incorporated.
Add the bread flour and knead for five minutes on medium speed or until the dough comes off the sides of the bowl and clumps around the dough hook.
Add the olive oil and knead. It will look like there is too much oil in the beginning, but don't worry. Be patient and the dough will eventually begin to absorb the oil. It took me about 25 minutes on medium-high speed for all of the oil to incorporate.
When all of the oil is gone and the dough looks smooth again, scrape it out of the bowl and onto a lightly floured board. Form into a smooth ball.
Place the dough into a large, oiled bowl (or back in the bowl of the mixer after oiling) and cover with cling wrap. Place in a warm place to rise for 90 minutes.
After 90 minutes, the dough should have doubled. Punch it down and once again form the dough into a smooth ball. Place it back in the bowl, cover, and refrigerate overnight or for 8-12 hours.
In the morning the dough should have risen again. If it's not doubled at least, let it stand outside in a warm place until it doubles. Otherwise, punch the dough down and shape into four even balls.
Let the balls stand on the countertop, covered with a kitchen towel, for 10 minutes. Meanwhile, oil a standard eight-inch loaf pan. Place the four balls side by side in the pan so they are touching each other.
If you want a shiny top, mix the remaining 2 tbsp milk with a tablespoon of aquafaba and some salt and apply to the top of the brioche. Do this once more just before putting the loaf in the oven.
Once the loaf has risen and domed over the pan (about 90 minutes), place in a preheated 375 degree oven and bake for 30 minutes.
Remove the loaf pan and let it cool on a rack until it can be handled. Remove the loaf from the pan and let it cool thoroughly on a rack. Serve.