Heat olive oil in a Dutch oven or heavy pot over medium heat; cook prosciutto until edges begin to curl and fat is rendered, 2 to 3 minutes. Add shallot and cook until fragrant, about 2 minutes. Add garlic and cook until fragrant, about 1 minute. Add mushrooms and cook for 30 seconds.
Cook and stir rice into prosciutto mixture, stirring continually, until rice is translucent around edges, 1 to 2 minutes. Pour red wine into rice mixture; cook, stirring every 30 seconds, until wine is absorbed, about 5 minutes. Stir 1 cup broth into rice mixture, cooking and stirring until broth is almost completely absorbed, 3 to 5 minutes. Continue adding 1 cup broth at a time, stirring constantly, until rice is tender, 15 to 20 minutes.
Mix arugula, Parmesan cheese, thyme, and black pepper into rice mixture; cook and stir until cheese is melted, 2 to 3 minutes.