Combine the flour, matcha powder, sugar and yeast in the bottom of the bowl of a stand mixer. Add the eggs and water, mixing with the dough hook until it comes together. With the mixer on low, add the salt, then the butter a spoonful at a time, mixing until it's incorporated into the dough. Mix on medium speed for 10 minutes until the dough is completely smooth.
Place the dough in a lightly oiled bowl, cover with plastic wrap and refrigerate. Leave in fridge for at least half a day, preferable overnight. If you want to speed up the process, let the dough double at room temperature then refrigerate for at least 30 minutes for the dough to firm up. This makes the shaping process considerably easier and less messy.
Make the filling:
Melt the butter and chocolate together until smooth. Stir in powdered sugar and cocoa powder; mixture should form a spreadable paste.
Assemble the loaf:
Coat a 9 x 4 inch loaf pan with oil or butter and line the bottom of each with a rectangle of parchment paper (which I didn't - oops). Retrieve dough from fridge and roll out into a rectangle of roughly 10 by 12 inches. Spread the chocolate mixture evenly over the dough, leaving a 1/2 inch border all around. Roll up the dough into a log and trim the last 1/2 inch off each end of the log. Slice 1/2 inch off each end of the log and slice it into half lengthwise. Lay the halves next to each other on the counter; pinch the top ends gently together. Lift one side over the other, forming a twist and trying to keep the cut sides facing out. Transfer into the prepared loaf pan.
Cover with a damp tea towel and leave to rise for another 1 to 1/2 hours at room temperature.
Preheat oven to 375F. Bake for roughly 30 minutes. If the babka browns too quickly halfway through baking cover with foil.
While the babka is baking, make the 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. Let the babka cool halfway in the pan then transfer to a cooling rack to cool the rest of the way before eating.