In a bowl, add dry ingredients: all-purpose flour, salt, sugar and yeast. Mix until combined.
Add wet ingredients: warm soy or oat milk and vegan butter. Using a stand mixer equipped with its dough hook, knead until dough forms, on medium-low. It might look there's too much vegan butter, but the dough will set after a minute or so.
Knead for 6 minutes more. Dough should feel smooth and tacky, and come out easily of the bowl.
Place dough into a clean bowl, covering it with a kitchen tower or plastic wrap. Let dough rise in a warm place for 90 minutes, or until doubled. It might take longer during a very cold winter, or less time during summer.
After it doubles, deflate dough, punching it down and removing from bowl.
Divide into 4 even pieces and let them rest covered for 10 minutes before shaping.
Prepare a 9 inch loaf pan, lining it with parchment paper.
In a clean surface, using a rolling pin or bottle, open dough, rolling it over itself and pinching to seal it close. Repeat with each piece, placing them side to side in the loaf pan, pinched side down.
Cover loosely and let it rise for about 40 minutes, or until it looks puffy and light.
While dough rises, preheat oven to 350°F, rack placed in the middle shelf.
Mix maple syrup and plant-based milk for the vegan wash. Bruch it on top of dough just before baking.
Bake for 35 to 40 minutes, or until a thermometer reads 180-190°F when placed in the middle of the brioche loaf.
Remove it from the loaf pan with the help of the parchment paper, and let bread cool on a rack.