Using a food processor, puree 6 tablespoons olive oil, the garlic and rosemary. Transfer to a large bowl and whisk in the vinegar. Transfer the steak and 1/2 cup marinade to a glass baking dish and mix. Add the radicchio to the remaining marinade and toss. Let stand for 1 hour.
Preheat a grill to medium. Arrange the radicchio on the grill in a single layer and cook until lightly charred, about 5 minutes. Season the steak with salt and pepper, arrange on the grill and cook, turning once, for 8 minutes for medium-rare. Let rest for 5 minutes, then thinly slice.
Brush the bread with 1 tablespoon olive oil and toast on the grill. In a large bowl, combine the arugula with the lemon juice and remaining 1 tablespoon olive oil; season with salt and pepper and toss. Layer each piece of toast with one-quarter each of the grilled radicchio and steak, then top with some of the arugula and shaved parmigiano-reggiano.