How to Make It
In a large pot of salted boiling water, cook the pasta just until al dente. Drain, reserving 1 cup of the cooking water.
In a large, deep skillet, heat the olive oil. Add the anchovies and garlic and cook over moderately high heat, stirring, until fragrant, about 1 minute. Add the broccoli florets and stems and cook for 2 minutes, stirring occasionally. Add 1/2 cup of the reserved pasta water, cover and cook until the broccoli is tender but still bright green, about 2 minutes. Mash the broccoli coarsely.
Add the pasta, Peppadews and the 1/2 cup of Parmigiano to the skillet and toss. Add the remaining 1/2 cup of pasta water and cook, tossing, until the pasta is coated with a light sauce, about 2 minutes. Stir in the parsley and serve in bowls, passing extra Parmigiano on the side.