Dexterous manipulation is limited by both control and design, without consensus as to what makes manipulators best for performing dexterous tasks. This raises a fundamental challenge: how should we design and control robot manipulators that are optimized for dexterity? We present a co-design framework that learns task-specific hand morphology and complementary dexterous control policies. The framework supports 1) an expansive morphology search space including joint, finger, and palm generation, 2) scalable evaluation across the wide design space via morphology-conditioned cross-embodied control, and 3) real-world fabrication with accessible components. We evaluate the approach across multiple dexterous tasks, including in-hand rotation with simulation and real deployment. Our framework enables an end-to-end pipeline that can design, train, fabricate, and deploy a new robotic hand in under 24 hours. The full framework will be open-sourced and available on our website: https://an-axolotl.github.io/HouseofDextra/ .