Lay the turkey breast skin-side down on a rimmed baking sheet. Sprinkle the meat with 1 tablespoon salt and 1 teaspoon pepper. Lightly cover with plastic wrap and chill until ready to assemble, at least 1 hour and up to 24 hours.
For the stuffing: Meanwhile, melt 1 tablespoon of the butter in a large skillet over medium-high heat. Add the sausage and cook, breaking up any large pieces with a wooden spoon, until lightly browned but not fully cooked through, 5 to 6 minutes. Transfer to a large bowl.
Return the skillet to medium-high heat and melt 2 tablespoons of the butter. Add the celery, onion and half of the thyme and garlic and cook, stirring occasionally and scraping up any brown bits, until tender and just beginning to brown, 10 to 12 minutes. Season with salt and pepper. Transfer to the bowl with the sausage.
Return the skillet to medium-high heat, pour in the wine and scrape up any brown bits. Simmer until the smell of alcohol subsides, about 2 minutes. Add the broth and bring to a simmer, then pour into the sausage mixture. Add the stuffing mix and stir to combine. Season with salt and pepper, cover and chill until ready to assemble.
For the mushrooms: Return the same skillet to medium-high heat and melt 2 tablespoons of butter. Add the mushrooms and cook, undisturbed, until they begin to brown in spots, about 3 minutes. Toss and cook on the other side, undisturbed, until browned in spots, about 3 minutes more. Add 1 tablespoon butter, the remaining garlic and thyme and some salt and pepper and continue to cook, tossing occasionally, until browned and very fragrant, about 3 minutes more. Transfer to a small bowl and let cool. Cover and chill until ready to assemble.
For the green beans: Meanwhile, bring a large pot of salted water to a boil and line a plate with a paper towel. Add the green beans to the boiling water and cook until bright green and crisp-tender, about 2 minutes. Scoop out with a slotted spoon and transfer to the lined plate. Let cool. Keep the water boiling.
For the squash: Line a plate with a paper towel. Add the squash to the boiling water and cook until tender but still holding their shape, 5 to 8 minutes. Transfer to the lined plate and let cool. Cover and chill until ready to assemble.
Position an oven rack in the lower third of the oven and preheat to 325 degrees F. Set a rack inside a rimmed baking sheet and coat the rack with cooking spray.
Spread the stuffing over the turkey breast, leaving a 1-inch border around the edges. Gently press the stuffing into the meat to adhere. Position the turkey so that one of the short sides is facing you. Starting 3 inches from the bottom edge, arrange a compact row of cranberries crosswise over the stuffing (maintaining the 1-inch border on the left and right sides). Arrange a compact row of green beans crosswise over the breast, next to but not covering the cranberries. Arrange the mushrooms crosswise over the breast, next to but not covering the green beans. Arrange the butternut squash crosswise over the breast, next to but not covering the mushrooms and ending no further than halfway down the length of the turkey.
Working from the bottom edge, roll the breast up tightly. Tuck in any filling that spills out of the sides once the turkey breast is rolled completely. Tie the roulade tightly with kitchen twine every 2 inches to make a compact log. Tie once more lengthwise to keep the filling secure. Place the roulade seam-side down on the prepared rack and baking sheet. Melt the remaining 2 tablespoons butter, then brush over the roulade. Sprinkle with salt and pepper.
Transfer to the oven and pour 6 cups warm water into the baking sheet. Roast until an instant-read thermometer inserted perpendicularly into the center of the thickest part of the roulade registers 150 degrees F, 3 to 3 1/2 hours (check on the skin after 2 hours and cover with foil once it is deep golden brown). Let sit at room temperature, loosely covered with foil, for 1 hour; the internal temperature will continue to climb and will reach 165 degrees F. Remove the twine, cut into 1-inch slices and serve with warm gravy.
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