In a small bowl, combine the active dry yeast, warm milk, and a pinch of the granulated sugar. Let it sit for 5-10 minutes until it becomes foamy.
In the bowl of a stand mixer fitted with the paddle attachment, combine the gluten free 1-to-1 baking flour, remaining granulated sugar, xanthan gum, and salt. Add the yeast mixture, eggs, softened butter, and vanilla extract. Beat on medium speed for 3-4 minutes until a soft, sticky dough forms. Cover the bowl tightly with plastic wrap and refrigerate for at least 2 hours or overnight.
For the filling, melt the semi-sweet chocolate and butter together in the microwave in 30-second intervals, stirring between each, until smooth. Whisk in the powdered sugar and cocoa powder. Let the filling cool until spreadable but not stiff.
Grease a 9x5-inch loaf pan and line it with parchment paper. On a large sheet of parchment paper generously dusted with gluten free flour, pat and roll the chilled dough into a roughly 10x16-inch rectangle. Spread the chocolate filling evenly over the dough, leaving a 1/2-inch border.
Starting from a long side, roll the dough up tightly into a log. Using a sharp, smooth edge, slice the log in half lengthwise to expose the chocolate layers. Twist the two halves around each other with the cut sides facing up, then carefully transfer the twist into the prepared loaf pan.
Cover the pan loosely with greased plastic wrap and let it rise in a warm spot for 45-60 minutes, until the dough has puffed up noticeably.
Preheat your oven to 350°F. Bake the babka for 40-45 minutes, tenting with foil after 25 minutes if the top is browning too quickly, until a toothpick inserted in the center comes out with just a few moist crumbs.
While the babka bakes, make the simple syrup by combining the sugar and water in a small saucepan. Bring to a simmer, stirring until the sugar dissolves. As soon as the babka comes out of the oven, brush the top generously with the warm simple syrup. Let it cool in the pan for 20 minutes before lifting it out onto a wire rack.