1If using charcoal grill, place drip pan directly under grilling area, and arrange coals around edge of firebox. Heat coals or gas grill for indirect heat.
2Rub cavity of turkey with salt and pepper; place onion and lemons in cavity. Brush oil over turkey. Insert barbecue meat thermometer so tip is in the thickest part of inside thigh muscle and does not touch bone.
3Mix butter and thyme; brush over turkey. Cover and grill turkey, breast side up, over drip pan or over unheated side of gas grill and 5 to 6 inches from medium heat 2 hours 30 minutes to 3 hours 30 minutes or until thermometer reads 165° and drumsticks move easily when lifted or twisted. If using charcoal grill, add about 15 briquettes every hour. Let turkey stand 15 minutes. Remove onion and lemons before carving.