Make the dough with standing mixer: add the sugar, yeast, egg, pumpkin purée and milk to the flour. Mix and knead until a smooth dough forms.
Cover the bowl with a plate (or plastic wrap). Let rise in a warm place until doubled in size, about 1-2 hours.
Make the filling: In a small bowl, stir the melted butter with sugar, pumpkin spice and cocoa powder until incorporated.
Grease a loaf pan.
On a lightly floured surface, roll the dough out into 0,2-0,3-cm (1/8-inch) thick.
Spread the filling evenly over the surface of the dough, than roll the dough up tightly. Cut the roll lengthwise in two. Twist the strands together and pinch at the end to seal. Carefully place the babka in one of the prepared pans.
Cover the loaf pan with plastic wrap. Let the babka rise in a warm place until doubled in size.
Preheat the oven to 190°C (375°F).
Bake the babka for 30 minutes.
Meanwhile, combine sugar and water in a medium saucepan over medium heat; bring to a boil, stirring, until sugar has dissolved.
After the babka is baked, pour the syrup over it.
Let cool slightly, then remove the babka from the loaf pan and let cool before slicing and serving.