Preheat the oven to 325 degrees F. Butter and flour a 12-cup Bundt pan; tap out the excess flour.
Heat 2 cups root beer, the cocoa powder, chocolate and 1 stick butter in a large saucepan over medium heat until the butter melts. Add the granulated and brown sugars and whisk until dissolved. Remove from the heat and let cool.
Combine 2 cups flour, the baking soda, allspice and 1 teaspoon salt in a bowl. Whisk the eggs into the root beer mixture, then gently fold in the dry ingredients (the batter will be slightly lumpy).
Pour the batter into the prepared pan and bake until a toothpick inserted into the cake comes out clean, 55 minutes to 1 hour, rotating the pan halfway through. Transfer to a rack. Gently poke the cake all over with a skewer and pour 1/4 cup root beer over the cake; let cool in the pan for 20 minutes, then invert the cake onto a serving plate and let cool completely.
Meanwhile, make the glaze: Whisk the remaining 1/4 cup root beer, a pinch of salt, the confectioners' sugar and vanilla in a bowl until smooth. Drizzle over the cake.