Pour milk, 1 tablespoon of the sugar, and yeast into a standing mixer fitted with a dough hook or paddle attachment. Allow to sit for 5 minutes. The mixture should foam, signaling that the yeast is active.
Add in the remaining sugar along with the vanilla bean paste, eggs, and flour. Turn mixer to low and allow to come together. Once mixture has just come together, transfer to a well floured surface and knead for 15 to 20 minutes or until the gluten is developed and the dough is less sticky. Note that it is a wetter dough, so this is OK!
Coat a large bowl with oil and place the dough into the bowl. Cover with plastic wrap and set in a warm area to proof for 1 hour.
for the filling
Meanwhile make the filling. In a small sauce pan, melt the butter and then add in the finely chopped chocolate, stirring to combine and melt. Add the sugar, cinnamon, and salt and mix well. Allow the mixture to cool completely before using.
Spray a 9 by 4 inch loaf pan with butter and flour, or parchment paper and set aside. Coat a large area very well with flour and turn the dough out onto the surface. The dough should have almost doubled in size. Roll out to about a 10-inch width (the side closest to you) and as long in length (away from you) as you can when rolling it thin, likely 10 to 12 inches.
Spread the chocolate and cinnamon mixture evenly over the dough, leaving a 1/2-inch border all around. Brush the end farthest away from you with water. Roll the dough up with the filling into a long, tight cigar. Seal the dampened end onto the log. I found that transferring the log to a lightly floured baking tray in the freezer for 10 to 15 minutes made it much, much easier to cut cleanly in half.
Trim last 1/2-inch off each end of log. Gently cut the log in half lengthwise and lay them next to each other on the counter, cut sides up. Pinch the top ends gently together. Lift one side over the next, forming a twist and trying to keep the cut sides facing out (because they’re pretty). Don’t worry if this step makes a mess, just transfer the twist as best as you can into the prepared loaf pan. In one batch, mine was long enough to “S” inside the pan and I nested the trimmed ends of the log in the openings. Even if you don’t (and choose to bake them separately in a little pan, as I did in other batches), the dough will fill in any gaps by the time it’s done rising and baking, so don’t worry.
Cover with a damp tea towel and leave to rise another 1 to 1 1/2 hours at room temperature.
Bake and finish the loaf. Heat oven to 375°F (190°C). Remove towel, place loaf on the middle rack of your oven. Bake for 30 minutes, but there’s no harm in checking for doneness at 25 minutes. A skewer inserted into an underbaked babka will feel stretchy/rubbery inside and may come back with dough on it. When fully baked, you’ll feel almost no resistance. If you babka needs more time, put it back, 5 minutes at a time then re-test. If it browns too quickly, you can cover it with foil.
While babka is baking, make syrup: Bring sugar and water to a simmer until sugar dissolves. Remove from heat and set aside to cool somewhat. As soon as the babka leaves the oven, brush the syrup all over it. It will seem like too much, but will taste just right — glossy and moist. Let cool about halfway in pan, then transfer to a cooling rack to cool the rest of the way before eating (an adorable suggestion from Ottolenghi — don’t worry, we know you’re going to eat it warm).
Do ahead: Babka will keep for a few days at room temperature. Longer, I’d freeze it. They freeze and defrost really well.