Combine the crushed garlic and salt to make a paste. Put the garlic paste into a small bowl and add the remaining paste ingredients.
Rub the paste evenly over both sides of each steak. Let the steaks stand at room temperature for 20 to 30 minutes before barbecuing. Meanwhile soak the broccoli.
Submerge the broccoli in a large bowl of cold water for 20 to 30 minutes. This will help them steam a little on the barbecue.
Prepare your barbecue for grilling (direct method) using the Weber Gourmet BBQ System Sear Grate, arranging the coals in the center of the barbecue, directly below the sear section of the grate.
Drain the broccoli and lightly drizzle with oil and season.
Place the steaks over the heat and close the lid. For a medium rare steak cook for 6 to 8 minutes, turning once (Adjust the time according to how you like your steaks cooked). Remove from the barbecue and allow to rest.
Using tongs, shake any excess oil from the broccoli. Place over the heat, with the lid closed as much as possible, until lightly charred, 3 to 4 minutes, turning occasionally. Move the broccoli over the in-direct heat for 2 to 3 minutes. Serve warm with the steaks.