Fill a large pot with about 3 quarts of water and bring to a boil. Salt generously (about 2 tablespoons), add pasta, and cook according to the package instructions. Reserve about 1 cup of pasta cooking water, then drain and return the pasta to the pot.
While the water comes to a boil/pasta cooks, make the capers and mix together the goat cheese mixture.
For the fried capers, heat the oil in a small skillet over medium-high heat. Once the oil is hot, add the capers and cook, stirring occasionally, until they darken in color, about 4-5 minutes. Stir in the garlic and red pepper flakes for about 1 more minute or until the garlic is just beginning to turn light golden brown (don’t burn it!). Use a slotted spoon or fork and transfer the capers/garlic onto a plate or bowl lined with a paper towel, leaving the oil behind (this oil is delicious for dipping bread or drizzle over a salad so don’t throw it away).
In a medium bowl, add goat cheese, olive oil, lemon zest (if using), pepper, and salt. Use a fork to mash and mix the oil into the cheese until smooth.
Add goat cheese mixture to the pasta along with ½ cup reserved pasta water and stir to coat. If it looks dry, add more pasta water and stir again. Taste and add more salt, if needed.
Stir in the arugula until it wilts down a little bit. Again, you’ll likely need to add more pasta water (as pasta sits, it absorbs liquid so you often have to add more water to loosen up the sauce).
Serve pasta topped with fried capers, parsley (if using), chopped nuts (if using), and parmesan cheese.