Pre-heat the oven to 375°F.
Spread the bread cubes out in a single layer on a sheet pan. Toast in the oven for 15 minutes and then transfer them to a large bowl.
Slice the delicata squash in half, lengthwise and scoop out the seeds. Brush the cut sides of the squash with olive oil and season with salt and freshly ground black pepper. Place the squash, cut side down, on the sheet pan and roast for 25 minutes. (You can toast the crumbs and roast the squash at the same time if both will fit in your oven.)
While the squash is roasting, make the stuffing. Remove the sausage from its casing, breaking it into chunks. Heat a large skillet over medium heat and add a little olive oil. Add the sausage and brown it, breaking it up into small crumbles. Transfer the browned sausage to the large bowl with the bread cubes using a slotted spoon.
Melt the butter in the skillet. Add the onions and celery and sauté for 2 to 3 minutes. Add the diced mushrooms, dried thyme and sage and sauté for a 6 to 8 minutes, until the vegetables soften. Season with salt and freshly ground black pepper. Add the vegetables to the bowl with the sausage and bread cubes, along with the chicken stock and cranberries. Toss to combine all the ingredients and to allow the bread cubes to absorb all the liquid. Stir in the beaten egg and fresh parsley.
Remove the squash from the oven and carefully turn it cut side up. Spoon the stuffing into the cavity of each squash half. Return the pan with the stuffed squash to the oven and bake for 20 to 25 minutes, until squash is fork tender and stuffing is browned on top.
Serve immediately with vegetables or a side salad.