Whisk the bread flour, sugar, yeast, and salt together in a large bowl or the bowl of a stand mixer. Add the lukewarm oat milk, vegan yogurt, and vanilla, then mix to form a rough, shaggy dough.

Knead on medium speed, or by hand on a lightly floured surface, for about 5 minutes until the dough comes together into a soft ball. Add the softened vegan butter a spoonful at a time and keep kneading for another 8 to 10 minutes, until the dough is smooth, elastic, and pulls away from the sides of the bowl.

Cover the bowl and let the dough rise at room temperature for 1 hour, until it has grown noticeably. Then move it to the fridge and let it cold-proof overnight, 8 to 12 hours, until doubled.

Make the filling while the dough is out. Melt the chopped vegan dark chocolate, vegan butter, cocoa powder, brown sugar, cinnamon, and salt together in a small pan over low heat, stirring until smooth. Take it off the heat and let it cool for about 20 minutes, until it is thick but still spreadable.

Punch down the chilled dough and roll it on a lightly floured surface into a rectangle about 12 by 16 inches (30 by 40 cm). Spread the cooled filling over the dough, leaving a 1-inch border on all sides, then scatter over the vegan chocolate chips and press them in gently.

Roll the dough up tightly from one long edge into a log. Place the log seam-side down on a tray and chill it in the fridge for 15 to 20 minutes so the filling sets before you cut it.

Use a sharp knife to cut the chilled log in half lengthwise, exposing the swirl. Turn the two halves cut-side up and twist them over each other into a loose braid, pinching the ends together. Lift the braid into a greased 9 by 5 inch loaf pan.

Cover the pan and let the babka proof at room temperature for 1 1/2 to 2 hours, until it looks puffy and has risen to near the top of the pan.

Heat the oven to 350F (175C). Bake the babka for 40 to 50 minutes, until the top is deep golden and an instant-read thermometer inserted into the center reads 195F (90C).

While the babka bakes, simmer the sugar and water in a small pan until the sugar dissolves, then set the syrup aside. Brush the hot loaf generously with the syrup as soon as it comes out of the oven. Cool it in the pan for 15 to 20 minutes, then turn it out onto a rack and let it cool completely before slicing.