For the brine: Combine the honey, salt, bay leaves, garlic, rosemary, thyme, red pepper flakes and 1 cup water in a large saucepot. Bring to a boil over medium-high heat. Turn off the heat and add the ice cubes. Place the turkey and brine into a 1-gallon resealable plastic bag. Refrigerate for 1 hour. Remove from the brine and pat dry with paper towels.
For the herb-roasted turkey breast: Preheat the oven to 425 degrees F.
Combine the garlic, parsley, rosemary, thyme, lemon zest, olive oil, salt and black pepper in a small mixing bowl and mix well. Rub the turkey all over with the herb-olive oil mixture to completely cover.
Place the turkey in a large cast-iron skillet and roast for 15 minutes. Reduce the heat to 300 degrees F and cook until the internal temperature reaches 160 degrees F, 75 minutes to 90 minutes. Cover with foil if the turkey starts to get too dark. Let the turkey rest for 10 minutes. Reserve all the juices from the bottom of the skillet.
Shred the turkey when cool enough to handle and reserve.
For the grilled cremini mushrooms: Preheat a grill to high. Add the creminis, olive oil, thyme, salt and pepper to a medium mixing bowl and toss well to coat. Place the creminis on the grill and char on all sides, 4 to 5 minutes. Reserve the mushrooms.
For the garlic butter: In a medium saucepan, combine the butter and garlic over medium heat and cook until the garlic becomes fragrant, 5 to 6 minutes. Add the parsley and mix well. Transfer to a small bowl and reserve.
For the sandwiches: Increase the oven to 425 degrees F. In the same pan the turkey was roasted in, heat the flour and 2 tablespoons of the butter over medium heat, whisking together to make a roux. Cook, whisking often, for 5 minutes. Stir in the wine, then add the cremini mushrooms and chicken broth. Bring to a boil, reduce to a simmer and cook until the gravy is reduced by half, 12 to 15 minutes.
Heat the remaining 2 tablespoons butter and shredded turkey in a large cast-iron pan over medium-high heat. Let the turkey crisp in the pan, 2 to 3 minutes. Lightly brush the rolls with the garlic butter and toast in the oven until golden brown, 6 to 8 minutes.
Spoon 1 tablespoon black truffle paste on a roll. Top with 2 ounces of the sliced truffle gouda laid from end to end, a quarter of the crisped turkey and some arugula. Spoon 2 tablespoons of the gravy over the roasted turkey. Make 3 more sandwiches with the remaining ingredients. Serve immediately with extra gravy on the side.