Heat the milk until it's a little bit hot to the touch. Whisk in yeast and 2 tsp sugar. Cover the bowl with a clean towel and let it proof for 5-10 minutes until the mixture becomes foamy (that's how you know your yeast is still active).
After the yeast has proofed, add remaining sugar, softened butter, vanilla extract and salt. Whisk it for 1-2 minutes until combined. Sift the flour and gradually add it in. Knead the dough until it starts pulling away from the sides of the bowl and looks soft and supple - about 7 minutes. If it looks too dry add a tablespoon of milk, or if it looks too wet add a little bit of flour at a time.
Form into a ball and place into a large bowl greased with butter, cover with plastic wrap and place it in a warm environment to rise until doubled, about 90 minutes or until it doubles in size.
After dough has risen, punch down the dough inside the mixing bowl. If you make both loaves, cover the bowl and place it in the refrigerator at least 4 hours, preferably overnight or up to 2 days. If you make only one loaf, cut the dough in half, the first one you refrigerate and the second one you place in the freezer.
Take out the dough from the refrigerator and let it rest at room temperature for 30 minutes. Grease two 10x30 cm loaf pans with butter and line with parchment paper.
Punch down the dough again, remove from the bowl and turn it out onto a lightly floured surface. Cut the dough in half and work with one loaf at a time, covering the second one with a clean towel.
Shape the babka - on a slightly floured work surface, roll one dough half out into a 20x30cm rectangle. Spread with 150g of Nutella, leaving a ½ cm border uncovered. Sprinkle 75g roughly chopped dark chocolate on top of the Nutella layer.
Tightly roll the dough into a log and place it on its seam. Using a sharp serrated knife, cut the log in half lengthwise. Place both halves with the chocolate filling side up and using a crisscross pattern, place one half on top of the other half, forming an X. Pinch the ends to secure the shape.
Place in prepared loaf pan and cover with plastic wrap or clean towel. Repeat with the second half of the dough.
Let both shaped loaves to rise for another 2 hours or until doubled in size.
Preheat oven to 180°C and bake for 40-45 minutes or until babka is golden brown on top and when you insert a skewer in the dough portion comes out clean.
While the loaves are in the oven, make the sugar syrup. In a small saucepan over medium heat, bring water and sugar to a boil, stirring constantly. Remove from heat when the sugar dissolves, let it cool slightly and add vanilla extract.
Remove loaves from the oven and brush syrup over them. Let them cool on a wire rack until they are warm, then remove from pans and serve warm or let cool completely.