Grate the carrots — you can use a box grater, food processor or a julienne peeler — then put them in a large bowl.
Set the oven broiler to high with a rack 6-inches from the heat source. Trim the root ends and ½-inch of the scallion tops. Line them up on rimmed sheet pan or piece of foil and place under the broiler. Broil about 5 minutes per side, until they're charred on the outer layer and soft inside. Slice into 1/2-inch pieces.
Whisk the lemon juice, salt and anchovies together in a small bowl to dissolve the salt. Pour over the carrots and toss. Grind 20 turns of the pepper grinder over the carrots and add the scallions, chili, parsley and walnuts. Toss well, then let sit for 10-15 minutes.
Pour the olive oil over and toss, then taste for seasoning, adding more salt, lemon juice and/or chili.
Tear the cheese into bite-size blobs and distribute over the salad.