In a large mixing bowl, combine sourdough starter, warm water, flour, sugar, salt, olive oil, and vanilla. Mix until a dough forms.
Knead until smooth, cover, and let rise overnight or about 8-10 hours at room temperature.
Roll the dough into a rectangle. Mix chocolate, cocoa powder, maple syrup, and melted oil into a spreadable paste.
Spread the chocolate mixture evenly over the dough, leaving a border. Roll up the dough tightly like a jelly roll.
Slice the roll lengthwise and twist the two halves together to form a braid.
Place the babka in a greased loaf pan, cover, and let rise another 2 hours.
Bake at 375°F (190°C) for 30–35 minutes or until golden brown.
Cool in the pan for 10 minutes, then transfer to a rack to cool completely before slicing.