Heat milk in a small saucepan until warm. Transfer to a large bowl and whisk in yeast; let sit until foamy, 5–10 minutes.

Whisk in egg yolks, vanilla, and 1/2 cup butter. Add sugar, salt, and 3 cups flour; mix until a shaggy dough forms. Knead dough on a lightly floured surface until supple, smooth, and no longer shiny, 5–10 minutes.

Transfer to a large buttered bowl. Cover and let sit in a warm place until doubled in size, 11/2–21/2 hours.

Mix chocolate, brown sugar, and cinnamon in a small bowl.

Turn out dough onto a lightly floured surface; divide into three portions. Shape each into a 12”-long rope. Roll out each rope to a 12x6” rectangle about ?” thick. Brush with butter and top with chocolate mixture, pressing gently. Roll up to form a log; pinch seam to seal (I always rubbed one edge of each seam with a bit of water before pinching to further insure it would not break apart).

Place logs, seam side down, side by side on a parchment-lined baking sheet. Pinch logs together at one end; braid, then pinch ends together and tuck under. Cover loosely and let sit in a warm place until 11/2 times larger, 1–2 hours.

Preheat oven to 350°. Beat egg yolk with 1 Tbsp. water in a small bowl. Brush dough with egg wash; sprinkle with granulated sugar. Bake until top is golden brown and “Babkallah” sounds hollow when bottom is tapped, 35–45 minutes. Let cool on a wire rack.