Preheat the oven to 325 degrees . In a medium skillet, heat 1 tablespoon olive oil over medium heat. Add the shallot and cook until softened. Add the bread cubes, prosciutto and mango. Stir in 1/2 cup chicken broth; remove from the heat and let cool. Stuff each hen with one quarter of the mixture and tie the legs together with kitchen string.
Sprinkle the flour on the hens. In a large Dutch oven, heat 2 tablespoons olive oil over medium heat. Add half the garlic and cook for 1 minute. Add the hens and cook, turning once, until browned. Arrange the hens breast side up, add the remaining chicken broth, cover and bring to a simmer. Transfer to the oven and cook until an instant-read thermometer registers 160 degrees when inserted in the thigh, about 55 minutes. Transfer the hens to a platter and tent with foil. Transfer the cooking liquid to a small bowl, reserving the Dutch oven. Let the liquid settle, then skim the fat from the surface; discard.
In the Dutch oven, heat the remaining 1 tablespoon olive oil and garlic over medium heat. Add the chard and cook until wilted, about 3 minutes. Serve alongside the hens with the reserved juices.