You can make the babka in a traditional 9-inch loaf pan or a wreath shape. For a wreath-shaped loaf, a 9-inch tube pan with a removable bottom is best. Whichever pan you choose, be sure to butter and line with parchment paper.
Put the dough on a floured surface and roll it into a 9-by-17-inch rectangle. Spread the caramelized onions evenly onto the dough, going all the way to the edge. Then break up the goat cheese with your fingers and layer it on top of the onions. As an optional bit of freshness, you can layer torn flat-leaf parsley now as well.
Then starting with a long side, roll into a tightly coiled log. Place log seam side down. Slice the log in half lengthwise to expose the filling. Twist the halves together as if you were braiding them.
Place the braided dough into the prepared pan. If you are using a loaf pan, let the shaped dough curl around itself if it’s a little too long for the pan. Then cover it loosely with plastic wrap, and let rise in a turned-off oven for 1 to 1 1/2 hours. The dough will puff up, but not quite double.
Heat oven to 375° F. Brush the top of the babka with oil. Bake until the top is deeply golden brown, 35 to 45 minutes. An instant-read thermometer inserted in the center will read 185 degrees. Transfer to a wire rack to cool slightly. The babka is best served still a little warm.