Combine the mushrooms, onion, cauliflower, thyme leaves, crushed red pepper, vinegar, and salt in a bowl. Toss well to combine, cover, and let sit at room temperature at least 1 hour, stirring once or twice. This will sort of cook the veggies and lightly pickle them.
Preheat the oven to 375 degrees F.
On a sheet tray lined with a silicone mat or parchment paper, make 4 flat circles of Parmigiano. Bake the Parmigiano disks until the cheese looks golden and lacy, 6 to 7 minutes. Remove the cheese rounds from the oven and let cool for 1 to 2 minutes, and then loosen them from the tray. Reserve.
Divide the braesola between 4 serving plates. Toss the mushroom mixture with the arugula and a drizzle of big fat finishing oil. Taste to make sure the mix is delicious. Arrange a pile of the veggie mix on top of the braesola on each plate. Crumble a Parmigiano disk on each plate.