6 oz parmesan, finely grated, divided
2 tbsp honey
½ small shallot, finely chopped
1/3 cup olive oil
3 tbsp white wine vinegar
1 tsp Dijon mustard
kosher salt and freshly ground black pepper
2 medium heads of radicchio, leaves separated, torn in half if large.
1 medium bunch arugula, tough stems removed
1 large pink lady apple, thinly sliced
Flakey sea salt such as Maldon
Preheat oven to 350 degrees. On a silicone mat lined baking sheet, divide 4 oz grated parmesan into 8 mounds. Press with your fingers to flatten. Bake until cheese is golden and melted, 6-8 minutes. Transfer baking sheet to a wire rack and let cool; break crisps into coarse pieces.
Heat honey in a small skillet over medium heat until warmed through. Whisk honey, shallot, oil, vinegar, and mustard in a large bowl; season with salt and pepper.
Add radicchio, arugula, apple, and remaining parmesan to vinaigrette; toss to coat. Season with salt and pepper. Top with crisps.
DO AHEAD: Vinaigrette and parmesan crisps can be made 1 day ahead. Cover and chill vinaigrette. Keep crisps at room temperature.