Combine water, 3/4 cup salt, and the sugar in a large stock pt, stirring until sugar dissolves. Add turkery, weighting it with a plate, if necessary, to submerge completely. 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 tsp salt with the side of a large knife until a paster forms. Transfer to a small bowl and add 4 tablespoons softened butter and herbs. Season with salt and pepper.
Using the handle for a wooden spoon, gently separate turkey skin from breast meat, being careful not to tear skin. Spread herb butter mixture evenly under the skin, then smooth down to remove trapped air. Brush turkey with 1 tablespoons melted butter and season with salt and pepper.
Make the veggies: Preheat oven to 425 degree with rack in lowest position. Place all the vegetables in a large bowl. Drizzle with oil, season with salt and pepper, and toss to combine.
Arrange veggies in a single layer on a rimmed baking sheet. Top with a rack. Place turkey breast on rack, skin skide up and roast for 30 minutes. Reduce oven temp to 325 degrees. Continue roasting, rotating sheet halfway through, and basting turkey with with pan juices and tossing veggies occasionally, until breast is golden brown and thickest part of the turkey registers 150 degrees on an instant-read thermometer, about 1 hour. Transfer veggies to a platter and turkey to a cutting board, and let rest for 30 minutes before carving.