In salad bowl (or do this in blender or food processor), using back of fork, mash together garlic and anchovy fillets into a paste. Whisk in mustard and egg yolk and then whisk in 1/2 cup oil a little at a time until thick sauce forms. Stir in lemon juice and season with fine sea salt and pepper.
Brush grill grate with oil. Coat romaine with remaining 2 tablespoons oil. Put romaine directly on grate over the fire and grill, turning once, just until grill-marked, about 20 seconds per side. Using tongs, remove romaine from heat and paint with half of the vinaigrette, getting the dressing down in between the leaves. Sprinkle with half of the cheese. Return romaine to the grill, in a spot with indirect heat and cook, covered, until cheese just starts to melt and the ends of the romaine leaves wilt, about 2 minutes.
Transfer romaine to platter. Dress with remaining vinaigrette, a scattering of remaining cheese, and a sprinkle of flake salt.
Nutrition information per serving:
Calories400Fat38 gSodium970 mg
Carbohydrates10 gSaturated fat6 gTotal sugars3 g