In a small bowl, whisk together the 2 tablespoons of finely grated Pecorino, lemon juice, a couple pinches of salt and a few grinds of pepper if you're using it. Slowly drizzle in the olive oil, whisking constantly. Taste for seasoning and set aside.
Slice the radishes into thin rounds. You should have about 1 cup.
In a serving bowl, combine the radishes, arugula, 3 tablespoons coarsely grated Pecorino and about two thirds of the dressing. Toss well, taste, and add more dressing if needed. Serve right away, before the radishes and arugula start to wilt!