In a large skillet, heat the oil over medium heat. Add the onions, leeks, celery, and garlic. Cook until the onions are translucent and softened, about 20 minutes, stirring often.
Add the mushrooms and cook until softened, about 5 minutes. Transfer the mixture to a large platter; spread out and let cool, about 5 minutes. (This can be refrigerated overnight).
In a large bowl, combine the onion-mushroom mixture, cubed bread, apricots, sage and thyme. Just before baking, pour the melted butter and broth over the mixture and stir gently to combine. Season with salt and pepper. Let stand for about 10 minutes, allowing the bread to soak up the liquid.
Preheat the oven to 375°F.
If you're stuffing your turkey, get as much as you can into the cavity of the turkey and put the rest in a greased baking dish. Bake until the top is crisp and the stuffing is heated through, 50 to 55 minutes. If just heating in a baking dish and not in the turkey, just add it to a baking dish or two and bake as directed.
Notes
If you've stuffed your turkey with stuffing, the stuffing should be removed when the turkey is done roasting. Place it in a baking dish and bake for about 30 minutes while turkey is resting (stuffing temperature should reach 165 degrees).
I cover my baking dish full of stuffing and refrigerate it until I'm ready to bake it. It turns out just fine that way.