Warm the plant-based milk until it reaches 100°F to 110°F (40°C to 45°C). Combine the warmed milk with the instant yeast and set aside for 5 minutes until the yeast becomes active and foamy.
In a large mixing bowl or the bowl of a stand mixer, combine the flour, granulated sugar, salt, vanilla powder or extract, and turmeric. Add the plant-based yogurt and the yeast mixture to the dry ingredients. Mix until a shaggy dough forms.
Add the room-temperature vegan butter to the dough. Knead the dough using a stand mixer fitted with a dough hook or by hand on a floured surface for 8-10 minutes, until it is smooth and elastic.
Place the dough in a lightly greased bowl, cover, and let it rise in a warm place for 1 hour, or until doubled in size.
While the dough rises, prepare the filling by melting the vegan butter, and chocolate chips in a double boiler. Once melted, remove from the heat and thoroughly whisk in the powdered sugar, cocoa powders, and vegan vegan yogurt until smooth. Chill for 30 minutes so that the filling firms up a bit.
Once the dough has risen, punch it down and roll it out on a lightly floured surface into a rectangle approximately 18 x 12 inches (45 x 30 cm).
Spread the chocolate filling evenly over the dough, leaving a ½-inch (1 cm) border on all sides. Starting with the longer edge, roll the dough tightly into a log, like you would for making cinnamon rolls. Pinch the seam to seal it to the body of the log.
Using a sharp knife, slice the log lengthwise into 2 equal halves. Position the halves with the cut sides facing up.
Twist the long ropes of babka dough together, keeping the cut sides exposed, to form a braid.
To bake as a loaf: Transfer the twisted dough into a greased 9 x 5-inch (23 x 13 cm) loaf pan, folding it as necessary to fit in your pan. Cover loosely and let it rise for 30-40 minutes, or until puffed and filling the pan.
Or, to bake as a wreath: Once braided, stretch the dough into one long, narrow braid. Connect the ends of the twisted dough to form a circle, pinching the ends together to seal. Carefully transfer the circle into a greased 9-inch (23 cm) round pan, ensuring the twist is evenly distributed. Cover loosely and let it rise for 30-40 minutes, or until puffed.
Preheat the oven to 350°F (175°C). Bake the babka for 40-45 minutes, or until the top is golden brown and a skewer inserted into the center comes out clean.
While the babka bakes, prepare the syrup by heating the water and brown sugar over medium heat until the sugar has dissolved. Remove from heat.
As soon as the babka comes out of the oven, brush the syrup generously over the surface. For the wreath version, ensure the syrup reaches all parts of the exposed surface. Allow the babka to cool in the pan for 10 minutes before transferring it to a wire rack to cool completely.