Arrange a roasting rack in a 9x13 baking pan and place turkey breast-side up on top. Set aside for a moment.
In a small bowl, combine crushed bay leaves, thyme, orange zest, salt, and brown sugar.
Use your fingers to loosen the skin on turkey and generously rub some of salt mixture under the skin and on the outside all over turkey. Refrigerate turkey uncovered at least 24 hours or up to 48.
Preheat oven to 350F. Remove turkey from refrigerator 30 minutes before you plan to roast it and use damp paper towels to rub dry brine off turkey.
In a small bowl, combine softened butter, garlic, and rosemary. Season butter with salt and pepper to taste.
Rub butter all over turkey breast, using your fingers to rub underneath the skin and on top. Scatter carrot, onion, and additional rosemary sprigs around turkey.
Bake turkey breast at 350F for 1 to 1-1/2 hours (about 20 minutes of cooking per pound of turkey) until turkey registers 160F when a kitchen thermometer is inserted in the thickest part of turkey (do not touch the bone if using bone-in turkey)
Carefully drain off the pan drippings into a medium skillet, lightly cover the turkey breast with foil, and allow to rest 20 minutes while you make the gravy.