Make the turkey: Combine water, 3/4 cup salt, and the sugar in a large stockpot, stirring until sugar dissolves. Add turkey, weighting it with a plate to submerge if necessary. Cover, and refrigerate for 6 hours.
Remove turkey from brine. Pat dry with paper towels, then let stand at room temperature for 1 hour.
Mash garlic and 1/2 teaspoon salt with the side of a large knife until a paste forms. Transfer to a small bowl, and add softened butter and herbs. Season with salt and pepper.
Using a chopstick or the handle of a wooden spoon, gently separate turkey skin from breast meat, being careful not to tear the skin. Spread herb-butter mixture evenly under skin, then smooth down to remove trapped air. Brush turkey with melted butter, and season with salt and pepper.
Make the vegetables: Preheat oven to 425 degrees with rack in lowest position. Place all of the vegetables in a large bowl. Drizzle with oil, season with salt and pepper, and toss to combine.
Arrange vegetables in a single layer on a rimmed baking sheet. Top with a rack. Place turkey on rack, skin side up, and roast for 30 minutes. Reduce temperature to 325 degrees. Continue roasting, rotating sheet halfway through, and basting turkey with pan juices and tossing vegetables occasionally, until breast is golden brown and the thickest part of the turkey registers 150 degrees on an instant-read thermometer, about 1 hour. Transfer vegetables to a platter and turkey to a cutting board, and let rest for 30 minutes before carving.
To carve the turkey breast: Slice each side along bone, then cut into slices at least 1/4 inch thick. Transfer to a serving platter.
Brine the turkey breast a day in advance. Pat it dry, cover tightly, and refrigerate overnight.