Cut off broccoli stems 1 inch below the florets. Starting at the stem end, slice each broccoli head lengthwise into 3 slabs.
In large baking dish, whisk together the EVOO, soy sauce and gingerroot. Coat the broccoli on both sides with the soy mixture. Let marinate at room temperature for about 1 hour.
Preheat a foil-lined grill to medium-high. Remove the broccoli from the marinade, shaking off excess liquid. Grill, covered, until lightly charred and crisp-tender, about 3 minutes per side.