On a cutting board, finely mince the garlic and anchovy fillets. Add the salt and, using the flat edge of your knife, mash everything into a smooth paste. Transfer to a small bowl.
Add mayonnaise, Dijon mustard, lemon juice, pepper, olive oil, and parmesan to the bowl. Whisk until smooth and creamy.
Brush both sides of each bread slice with olive oil. Grill over medium heat for about 3 minutes per side, until crispy. Let cool, then cut into cubes.
Halve the romaine hearts lengthwise. Brush all sides with olive oil and sprinkle with a pinch of salt. Grill for 1–2 minutes per side, until lightly charred. Remove and cool slightly. Cut crosswise into 1-inch strips.
In a large bowl, combine grilled romaine strips, bread cubes, and grated parmesan. Add dressing and toss gently to coat. Serve immediately.