Bring a large pot of water to a rolling boil over high heat. Add 1 teaspoon salt, then cook the penne according to package directions until al dente. Reserve 1 cup of the pasta cooking water, then drain the pasta.
While the water heats, heat 2 tablespoons extra-virgin olive oil in a large saucepan or deep skillet over medium heat. Add the crushed garlic and chopped anchovy fillets (if using) and sauté until the garlic is fragrant and the anchovies begin to dissolve, about 2–3 minutes. Remove and discard the garlic.
Add the halved or quartered tomatoes to the pan and stir to coat in the flavored oil. If using anchovy paste instead of fillets, add it now. Cook, stirring occasionally, until the tomatoes release their juices and the sauce thickens slightly, about 10 minutes.
Add the arugula and 1/2 teaspoon black pepper to the tomato sauce and toss until the greens are just wilted, about 1–2 minutes.
Add the drained pasta to the pan and toss with the sauce and 4 tablespoons grated Pecorino. If the sauce seems dry, add a few tablespoons of the reserved pasta water until you reach the desired consistency. Taste and adjust seasoning if needed.
Turn off the heat. Serve the pasta drizzled with a little extra-virgin olive oil and sprinkled with additional Pecorino if desired.