Season the cavity of each hen with salt and pepper. In a medium skillet over medium heat, melt 1 teaspoon of the butter. Add the mushrooms and season with salt and a few grinds of pepper. Add the thyme and cook, stirring occasionally, until the mushrooms are soft and the juices have evaporated, 5 to 8 minutes. Transfer to a bowl to cool. Stir in the prosciutto and truffle oil.

Heat the oven to 425 degrees F. Insert your fingertips at the wing end of the breast and gently loosen the skin over the breast and around the legs, being careful not to tear the skin. Divide the stuffing into two equal portions of about 2 tablespoons each. Place one portion of the stuffing under the skin and with your fingers, distribute it evenly over the breast and thigh. With kitchen twine, tie the legs together. Tuck the wings underneath. Repeat with the other hen.

Arrange the birds breast side up on a wire rack set in a shallow roasting pan. Melt the remaining 2 teaspoon butter and use half to brush over the hens. Season each hen with salt and pepper. Roast for 20 minutes and brush with the remaining melted butter. Roast until the juices run clear when you prick the thickest part of the thigh and an instant-read thermometer inserted in the thigh registers 170 degrees F, another 25 to 30 minutes. Transfer the hens to a platter, tent with foil, and let stand for 10 minutes before serving.