PREPARATION:

1. Cover mesquite chips with cold water; soak 20 minutes. Prepare grill for indirect cooking. Rinse turkey; pat dry with paper towels. Place onion, lemon and 3 tarragon sprigs in cavity. Pull skin over neck; secure with metal skewer. Tuck wing tips under back; tie legs together with wet kitchen string.
2. Spread softened butter over turkey skin; sprinkle with salt and pepper. Insert meat thermometer into center of thickest part of thigh, not touching bone.
3. Drain mesquite chips; sprinkle 1 cup over coals. Place turkey, breast side up, on grid directly over drip pan. Grill turkey, on covered grill, over medium coals 11 to 14 minutes per pound, adding 4 to 9 briquets to both sides of fire each hour to maintain medium coals and adding remaining 1 cup mesquite chips after 1 hour of grilling. Meanwhile, soak additional fresh tarragon sprigs in water.
4. Combine melted butter, lemon juice, chopped tarragon and garlic in small bowl. Brush half of mixture over turkey during last 30 minutes of grilling. Place soaked tarragon sprigs directly on coals. Continue to grill, covered, 20 minutes. Brush with remaining mixture. Continue to grill, covered, about 10 minutes or until thermometer registers 180°F.
5. Transfer turkey to carving board; tent with foil. Let stand 15 minutes before carving. Discard onion, lemon and tarragon sprigs from cavity.