Preheat oven to 350°. Put radicchio on a large rimmed baking sheet. Lightly brush all over with oil, sprinkle with salt and pepper, and bake until wilted, 8 to 10 minutes. Meanwhile, open up bread on another baking sheet and bake until lightly toasted, 8 minutes. Let radicchio and bread cool.
Layer radicchio on bottom half of ciabatta, then top with roasted peppers, provolone, salami, and basil. Close bread and cut into 4 sandwiches. Combine vinegar and mustard in a small bowl. Just before serving, drizzle dressing into sandwiches through a cut side. Serve with plenty of napkins.
Keeps: Up to 2 hours at room temperature, 3 hours with ice packs.
Note: Nutritional analysis is per sandwich.