In a small saucepan, melt the butter and saute the shallots over medium heat until translucent, about 5 minutes. Add the apricot preserves, Dijon mustard, and soy sauce and simmer until thick, about 10 minutes. Season with salt and pepper and set aside to cool.
Season the game hens liberally with salt and pepper. Using a basting brush, paint each bird entirely with the glaze.
Place the glazed birds in a roasting pan, breast side up. Roast for 20 minutes, uncovered, and then reglaze the birds. Repeat this process twice more, every 15 minutes, at 35 and 50 minutes. If the hens are turning too brown, lightly tent the roasting pan with foil to protect them. After 60 minutes, check with an instant-read thermometer; the thighs must register at 165 degrees F, the breasts at 170. Let rest for 10 minutes and serve.