Trim fat from steak. Place steak in a plastic bag set in a shallow dish. For marinade, in a small bowl combine the 1/4 cup pineapple juice, the cooking oil, lemon juice, the 1 teaspoon ginger, and the mustard. Pour over steak; seal bag. Marinate in the refrigerator for 4 to 24 hours, turning bag occasionally.
At least 1 hour before smoke cooking, soak wood chips in enough water to cover. Drain before using. Drain steak, reserving marinade.
In a charcoal grill, arrange medium-hot coals around a drip pan. Test for medium heat above the pan. Sprinkle the drained wood chips over the coals. Place steak on the grill rack over drip pan. Cover and smoke until steak is desired doneness, brushing once with marinade halfway through cooking. [Allow 22 to 26 minutes for medium-rare (145 degrees F) or 26 to 30 minutes for medium doneness.] [In a gas grill, preheat grill. Reduce heat to medium (160 degrees F). Adjust for indirect cooking. Smoke as above, except add drained wood chips according to the manufacturer's directions.] Discard the remaining marinade.
Meanwhile, for sauce, in a small saucepan heat peanut butter over low heat just until melted. Gradually stir in the 1/4 to 1/3 cup pineapple juice until creamy (peanut butter mixture may thicken at first but will become creamy as more juice is added). Remove from heat. Stir in jalapeno pepper, green onion, and, if desired, the 1/2 teaspoon ginger.
To serve, thinly slice steak across the grain. Serve steak with sauce. If desired, garnish with additional green onion. Makes 8 servings.