Simmer the water, salt, brown sugar, garlic, bay leaves, thyme, and oregano for 45 min. Cool, add the ice, chill the brine completely to 40°F or below.
Submerge the fully thawed turkey breast-side down in the chilled brine. Cover; refrigerate at 40°F or below for 24–36 hr.
Discard brine and thoroughly pat the turkey dry; do not rinse.
Stabilize the board with a damp towel; cut out the backbone and reserve it. Turn the turkey over and press the breastbone to flatten it.
Heat smoker to 300°F. Set turkey breast-side up on the grate and smoke for an estimated 2½–3½ hr, monitoring temperature.
Stir together the optional glaze ingredients. Brush on during the last approximately 30 min of cooking.
Cook until the thickest breast, innermost thigh and innermost wing each reach at least 165°F, measured away from bone.
Rest approximately 20 min, carve and serve.