4 medium carrots, halved length-wise, 1 inch of greens attached
4 small turnips, peeled and halved
1 1/2 Lbs. assorted small potatoes, scrubbed, halved or quartered
2 Tbsp EVOO
Brine turkey: Combine water, 3/4 cup salt, and the sugar in a large stock pot, sturring until salt and sugar dissolve. Add turkey, weighting it down with a plate if necessary. Cover and refigerate for 6 hours.
Remove turkey from brine. Pat dry with paper towels, then let stand at room tempterature for 1 hour.
Mash garlic and 1/2 Tsp. 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 the breast meat, being careful not totear 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 begetables: Preheat oven to 425º with rack in lowest position. Place all of the vegetables in a large bowl. Drizzle with oil, season with salt and pepper, 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 325º. Continue roasting, rotating sheet halfway through, and tossing vegetables occasionally, until breast is golden brown and the thickest part of the trukey registers 150º, 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