Cook first 4 ingredients in medium saucepan over low heat until garlic and ginger begin to brown, about 8 minutes. Set aside. DO AHEAD Sauce can be made 1 day ahead. Cover; chill. Rewarm before using.
Prepare barbecue (medium-high heat). Drop 1 lobster headfirst into large pot of boiling water. Cover; cook 3 minutes (lobster will not be fully cooked). Using tongs, transfer lobster to baking sheet. Return water to boil. Repeat with remaining lobsters.
Transfer 1 lobster, shell side down, to work surface. Place tip of large knife in center of lobster. Cut lobster lengthwise in half from center to end of head (knife might not cut through shell), then cut in half from center to end of tail. Use poultry shears to cut through shell if necessary. Repeat with remaining lobsters.
Place lobster halves, meat side up, on barbecue. Brush meat with some basting sauce; sprinkle with salt and pepper. Cover barbecue; grill lobsters until just opaque in thickest portion of tail, about 9 minutes. Remove from barbecue; arrange lobsters on platter. Brush with more sauce and sprinkle with green onions. Serve, passing remaining sauce separately.