Step 1 In a mortar and pestle or a blender, puree the anchovy fillets, garlic and salt. Slowly add the olive oil in a stream, stirring or pureeing constantly to make a creamy sauce. Stir in the lemon juice, taste and add more lemon juice or salt if necessary. Set aside.
Step 2 Prepare a wood fire or heat a grill pan over medium-high heat until hot. Split the hearts of romaine in half lengthwise, leaving them attached at the base. Brush both sides of each half with the anchovy sauce and then grill over a wood fire or in a grill pan until the leaves sizzle and begin to look a little frazzled, about 2 minutes on each side. They probably won't brown much.
Step 3Remove the cooked romaine to individual serving plates, cut-side up. Sprinkle with chopped walnuts and spoon over some more dressing. Use a vegetable peeler to shave Parmigiano-Reggiano in thin sheets over top. Serve immediately.