Preheat oven to 350 degrees F.
Slice off the top part of garlic bulbs so tops of cloves are exposed; place garlic bulb in an oven-safe ramekin. Pour 2 TBSP olive oil over garlic bulb. Cover ramekin with aluminum foil. Bake in the preheated oven until cloves are soft and browned, 30-45 minutes. Set aside to cool.
Place Brussels sprouts, prosciutto, and cranberries in a 9×13-inch baking dish. Pour in 3 TBSP olive oil, chicken broth, and balsamic vinegar; toss to coat. Season with salt and black pepper. Squeeze the softened garlic cloves into a small bowl and mash slightly. Add mashed garlic to Brussels sprouts mixture, mix well.
Bake in the preheated oven until sprouts are browned, about 30 minutes.