Cook butter, anchovies, garlic, and pepper flakes in small saucepan over medium heat, stirring frequently until garlic turns straw-colored, about 4 minutes. Off heat, whisk in lemon juice. Cover to keep warm and set aside.

Pat beef dry with paper towels and sprinkle with 1 teaspoon salt and ½ teaspoon pepper. Thread beef onto four 12-inch metal skewers. Whisk 3 tablespoons oil, water,¾ teaspoon salt, and remaining½ teaspoon pepper together in large bowl. Add broccoli wedges to oil mixture and toss to coat. Push toothpick horizontally through each onion round to keep rings intact while grilling. Brush onion rounds with remaining 1 tablespoon oil and sprin­kle with remaining ¼ teaspoon salt.

For a charcoal grill: Open bottom vent completely. Light large chimney starter filled with charcoal briquettes (6 quarts). When top coals are partially covered with ash, pour evenly over grill. Set cooking grate in place, cover, and open lid vent completely. Heat grill until hot, about 5 minutes.

For a gas grill: Turn all burners to high; cover; and heat grill until hot, about 15 minutes. Leave all burners on high.

Clean and oil cooking grate. Grill beef, broccoli, and onion until meat is charred and registers 120 to 125 degrees (for medium-rare) and vegeta­bles are crisp-tender and browned, 10 to 12 minutes, flipping as needed. Remove toothpicks from onion rounds. Sprinkle beef and vegetables with chives. Serve with butter sauce.