Preheat the oven to 400 degrees F.
In a medium skillet, melt the butter over medium-high heat. Add the shallots and garlic and cook, stirring, for 30 seconds. Add the mushrooms, 1/4 teaspoon of the salt, and white pepper, reduce the heat to medium, and cook, stirring, until all the liquid has evaporated and the mushrooms begin to caramelize, about 12 minutes. Add the wine and cook, stirring to deglaze the pan, until all the liquid has evaporated. Remove from the heat and let cool before using.
With your fingers, gently lift and loosen the skin from the hens' breasts to make a pocket. Spread about 2 tablespoons of the mushroom duxelle over each breast and pull the skin back into place. Season the hens, inside and out, with the remaining 3/4 teaspoon salt and the black pepper. Wrap each hen with 2 strips of bacon, crossing over the breast to cover.
Place the hens breast side up in a roasting pan and roast until the juices run clear and an instant-read thermometer inserted into the thickest part of the thigh registers 180 degrees F, about 45 minutes. Remove from the oven and serve.