A squeeze of lemon lends this dish a bright flavor.
Keeping base intact, quarter radicchio so head opens out into four wedges. Combine vinegar, salt, pepper, and 1 1/2 teaspoons oregano. Whisk in oil. Brush half the vinaigrette over and into cut radicchio.
Place a grill pan over medium heat. Place radicchio, cut-side down, on the pan. Cover radicchio with a metal bowl. Cook 2 to 4 minutes, until leaves start to soften. Turn radicchio over. Tuck some of the grated cheese into leaves, separating layers a little; sprinkle some more cheese over surface, leaving a little for garnish. Cover, and cook on low heat until outer leaves are soft and cheese has melted, 3 to 5 minutes. Transfer to serving platter, drizzle with remaining half of the vinaigrette, 1/2 teaspoon oregano, and cheese. Serve with lemon wedges.