Melt your butter in a small saucepan. Add the non-dairy milk and water, then return the saucepan to low heat. Heat the liquid to 100-115F (or until it's hot to the touch when you dip your pinkie finger in, but not painful).
In a large mixing bowl, whisk together the flour, yeast, sugar, salt, mahlep, and mastic.
Add the aquafaba to your liquid mixture, stir, and then add this liquid to your dry ingredients. Use a spatula to mix until a sticky dough ball forms.
Turn your dough onto a lightly floured surface. Knead for 10-14 minutes, or until your dough is smooth and pliable. If it's very sticky as you knead, you can add a little extra flour to your surface, but try not to add a lot. This is a sticky dough in general, so use a bench scraper to help you as you knead. Alternately, you can mix your dry ingredients in the bowl of a stand mixer fitted with the dough hook, add your liquid ingredients, and knead on medium high in the stand mixer for about 6-7 minutes. Once again, the dough is ready when it's smooth and supple.
Transfer the dough to an oiled bucket or bowl and turn once to coat. Cover and allow the dough to rise for 1-2 hours, or until doubled in size (this will depend on the temperature in your home).
Preheat your oven to 350F. Transfer the dough onto a lightly oiled surface and cut it into three even pieces (they'll be about 285-300 g each). Roll these into strands 19-20 inches long. Arrange the strands lengthwise on a rectangle of parchment. The short edge of the parchment (and ends of the strands) should be facing you. Braid the strands into a 3 strand plait, tucking in the ends after you're finished. Transfer the parchment to a baking sheet. Allow the tsoureki to proof for 45-75 minutes, or until puffy. When you poke it gently with your finger, it should spring back, but slowly.
When the dough has finished proofing, combine the non-dairy milk and agave or maple syrup in a small bowl. Brush the dough with this vegan version of an egg wash. Transfer the dough to your oven and bake for 20 minutes. Remove the dough from the oven and brush it with another layer of the egg wash, then sprinkle it with sesame seeds. Return it to the oven in a rotated position (so that it bakes evenly), and then bake it for another 20-30 minutes, till deep golden brown throughout (a probe thermometer should read at least 190F in the center of the bread). Allow it to cool on a cooling rack before slicing and serving.