Step 1Trim the ends of the radicchio and cut in half lengthwise, then cut the pieces into quarters. Finely shred each piece.
Step 2Heat the oil in a large skillet. Add the garlic and cook 1 minute. Add the radicchio and cook, stirring frequently, until wilted, 4 minutes. Sprinkle the salt over top and stir well. Set aside to cool.
Step 3Roll the puff pastry out on a lightly floured surface into a rectangle about 14 by 12 inches. Lay the radicchio in a stripe down the center. Dot the top of the radicchio with the cheese. Fold both sides of the pastry over the center and brush the underside edge with the egg wash; press to seal. Brush the rest of the egg wash over the surface of the strudel. Place the strudel on a baking sheet and refrigerate 10 to 15 minutes.
Step 4Meanwhile, heat the oven to 400 degrees.
Step 5Bake until golden brown, 20 to 25 minutes. Cut crosswise into 12 pieces and serve.