For the turkey breast: Prepare the turkey breasts for brining 1 day in advance. Bring 1 gallon of water to a boil in a large pot and whisk in the full jar of turkey brine mix. Cook, stirring, until the brine mix is dissolved then remove from the heat. Add 2 gallons of ice and stir until dissolved and the liquid is cool to the touch. Put the turkey breasts in the brine, cover and refrigerate for 12 to 24 hours.
Remove the turkey breasts from the brine and place in a colander. Rinse thoroughly, then transfer to a paper-towel-lined baking sheet. Dry both turkey breasts with paper towels and use your fingers to create a pocket between the breast meat and skin. Coat each breast with the softened butter, being sure to push the butter into the pocket underneath the skin.
Preheat the oven to 350 degrees F. Drizzle 3 tablespoons of the oil onto a baking sheet.
Put the carrots, garlic, celery, lemon and onion on the baking sheet. Sprinkle generously with salt and pepper. Toss the vegetables to coat with the oil and spread in an even layer. Put the rosemary, thyme and sage on top of the vegetables and season with 1 tablespoon of the poultry seasoning. Put both turkey breasts on top skin-side up and season with the remaining poultry seasoning. Drizzle the remaining 1 tablespoon oil over the turkey breasts and roast until the skin is crisp and an instant-read thermometer inserted in the thickest part of the breasts reads 165 degrees F, about 1 hour.
Allow the roasted turkey to rest on the baking sheet for 15 to 20 minutes. Transfer to a cutting board. Pour the juices remaining in the baking sheet into a fat separator for use in the turkey gravy.
For the biscuits: Separate the dough into biscuits and arrange them on a parchment-lined baking sheet. Brush the top of each biscuit with melted butter and sprinkle each with 1 tablespoon grated Cheddar. Bake according to the package directions until golden brown. Set aside and keep warm.
For the home fries: Put the frozen potato chunks on a baking sheet. Drizzle with the oil and sprinkle with salt and pepper. Toss to coat the potatoes and roast according to the package directions until golden brown and crispy. Set aside and keep warm.
For the turkey gravy: Put the turkey gravy in a small saucepan over medium heat. Whisk in the chicken stock and the defatted liquid from the baking sheet. Cook about 3 minutes. Add the Cheddar and cook, stirring, until melted and smooth, about 2 minutes more. Stir in the dried sage and a pinch of black pepper. Set aside and keep warm.
For the cranberry relish: Stir together the cranberry sauce, lemon zest, cinnamon, a pinch of salt and 1/4 teaspoon pepper in a medium bowl.
To assemble the sandwiches: Slice the turkey breasts into 1/4- to 3/8-inch-thick slices and halve each biscuit. Melt the butter in a large nonstick skillet over medium-low heat. Working in batches, fry the eggs until the whites are set and the yolks are runny, about 3 minutes each. Sprinkle each egg with a pinch of salt and black pepper.
Place a heaping tablespoon of cranberry sauce on the bottom half of each biscuit. Top the cranberry sauce with 1/3 cup home fries. Top the home fries with 3 slices of roasted turkey breast. Drizzle each sandwich with 2 tablespoons of warm gravy and top with more cranberry sauce. Gently place a fried egg on top of each and sprinkle with the parsley. Top each sandwich with the top half of the biscuits and serve immediately.
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