Thinly slice the radishes and florets of cauliflower on a mandoline or with a vegetable peeler. Cover with a damp towel and reserve. Wash and dry the radish tops if using.
Cut the pomegranate in half. Save one half for another use. Cup the pomegranate half in your hand, hold over a large bowl and beat it with the back of a spoon to expel the seeds. The bowl should catch the seeds and any juice.
Add the lemon zest and juice, 1 teaspoon kosher salt and sugar to the bowl and whisk in the olive oil.
When ready to serve, add the arugula (and radish tops if using) to the bowl with the dressing and toss gently to combine. Season to taste with salt and pepper. Add the shaved radishes and cauliflower and gently mix. Scatter the cheese over the top of the salad and serve immediately.
Make a 1-inch slice in the pomegranate and then pull open to reveal the seeds. This will prevent you from cutting most of the seeds before you expel them.