Using a fine grater, grate one garlic clove into a medium mixing bowl. Add the finely grated Parmesan cheese, mayonnaise, olive oil, lemon juice, Dijon mustard, and fish sauce. Whisk all the ingredients together until the dressing is smooth and emulsified. Season the mixture with salt and freshly ground black pepper to your preference.

Preheat a grill to medium-high heat and allow it to warm for about 5 minutes. Alternatively, heat a grill pan over medium-high flame. Lightly brush both sides of the sourdough slices and the cut sides of the romaine lettuce with olive oil. Sprinkle each with salt and freshly ground pepper to enhance flavor.

Place the bread on the hot grill and cook until deeply charred and crispy, about 1 to 2 minutes per side. Remove and transfer to a serving plate. Meanwhile, grill the romaine halves cut-side down until they develop a light char and are slightly wilted, approximately 1 to 2 minutes.

Once the bread is grilled, rub one side of each slice with the remaining garlic clove to infuse it with a subtle garlic aroma. Tear the grilled bread into rough 1/2-inch pieces to create rustic-style croutons.

Spoon some of the prepared dressing onto a serving platter and arrange the grilled romaine on top. Drizzle additional dressing over the lettuce, then scatter the croutons across the dish. Finish with a generous sprinkle of shaved Parmesan for a final savory touch.