Mix the flour with 1/4 teaspoon each salt and pepper in a large bowl. Add the artichoke hearts and toss. Heat the olive oil in a large skillet over high heat. Add the artichoke hearts and fry until golden brown and crisp, about 3 minutes per side. Transfer to a paper towel-lined plate to drain. Remove the skillet from the heat and reserve the frying oil.
Season the tomatoes with salt and pepper. Divide the mozzarella, tomatoes, prosciutto and fried artichokes among plates. Top with the basil and olives and drizzle with some of the reserved frying oil.