Step 1

Whisk together Parmesan, mayonnaise, anchovy paste, lemon juice, balsamic vinegar, garlic, Worcestershire sauce, and Dijon in a large bowl; season with salt and pepper to taste. Set aside, or cover and refrigerate until ready to use.

Step 2

Prepare a charcoal or gas grill for two-zone grilling: Open bottom vent of a charcoal grill completely. Light a charcoal chimney starter filled with briquettes. When briquettes are covered with gray ash, pour them onto bottom grate of grill. Push two-thirds of coals to one side of the grill, leaving one-third of coals on the opposite side. If using a gas grill, preheat to medium-high (400°F to 450°F) on one side and medium-low (300°F to 350°F) on the other side.

Step 3

Brush lettuce, spring onions, radishes, fennel bulbs, and mushrooms with grapeseed oil, and season liberally with salt and pepper. Place lettuce on unoiled grates over side of grill with fewer coals or the medium-low side of gas grill. Grill, uncovered, until outer leaves are lightly charred and inner leaves are tender, 2 to 3 minutes per side. Meanwhile, working in batches if necessary, place spring onions, radishes, fennel bulbs, and mushrooms on unoiled grates over side of grill with more coals or the medium-high side of gas grill, and grill, uncovered, turning occasionally, until lightly caramelized and tender, 6 to 7 minutes.

Step 4

Cut lettuce and fennel bulbs into 2-inch pieces; discard cores. Cut spring onions into 2-inch pieces. Add all vegetables to bowl with dressing, and toss to coat. Divide salad evenly among 4 plates, and top with additional Parmesan, lemon juice, and salt to taste. Serve warm.