In a saucepan add your milk (or water) and vegan butter over medium heat, stirring occasionally until the butter has melted. Set aside.
In your bowl with the yeast add your sugar and brown sugar (or honey) and salt. Ensuring the milk and butter mixture isn't too hot, add to the bowl and stir to combine. Now add your flour, 1 cup at a time, until the dough is still a little sticky but can be handled.
Knead for 8 minutes, and then place the dough in a greased bowl and covered with a clean towel in a warm place to proof for 40 minutes.
After 40 minutes, punch down the dough which should have doubled in size. Divide into pieces using a kitchen scale. For two three-strand loaves, divide into six, for two four-strand loaves into eight, for two six strand loaves into 12. For hamburger sized buns divide into 20 pieces of about 100g each, for dinner rolls, divide into 36 pieces of about 50g each.
Roll each piece into a rope and either braid the loaf or tie a knot. Place onto a lined baking sheet or baking stone, and cover with a clean towel for 60 minutes.
As your second proof is finishing, preheat your oven to 350°F (175°C). Using the last egg, use a pastry brush to glaze the dough with the egg wash. If desired, add sesame, nigella, or poppy seeds, or za'atar to garnish.
Cooking time for loaves is roughly 40-50 minutes, or until a tap on the bottom makes a hollow sound and the tops are golden brown. Rolls take between 15 to 20 minutes depending on size.