Position a rack in the lower third of the oven and heat the oven to 425°F. Bring a large pot of well-salted water to a boil.
Toss the cauliflower, tomatoes, oil, 3/4 tsp. salt, and 1/2 tsp. pepper on a rimmed baking sheet; spread in a single layer. Roast, stirring once or twice, until the cauliflower begins to turn golden and tender, about 15 minutes.
Meanwhile, pulse the sage and garlic in a food processor until minced. Add the prosciutto and pulse until coarsely chopped. Once the cauliflower is golden, toss the herb mixture into the vegetables and continue to roast until fragrant and the cauliflower is golden brown, 5 to 7 minutes.
Boil the orecchiette until al dente, 9 to 10 minutes. Reserve 1 cup pasta-cooking water. Drain the pasta and return it to the pot. Stir in the roasted cauliflower mixture, arugula, cheese, and enough pasta water to moisten. Season to taste with salt and pepper.
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