1/2 cup oil-packed sun-dried tomatoes, drained and chopped
2 to 3 ounces goat cheese, crumbled while still cold (around 1/2 cup)
1/3 cup coarsely grated Parmigiano-Reggiano (or Parmesan)
15 pitted kalamata olives, chopped (optional)
1/2 small lemon, juiced
optional add-ins: 2 cups or 1 can cooked chickpeas, drained, and/or a couple handfuls of baby arugula (great with leftovers)
Bring a large pot of salted water to a boil. Place a colander in the sink with a glass liquid measuring cup or heat-safe bowl next to it. Add the pasta to the boiling water and cook until al dente, as directed on the package instructions. Remove the pot from heat and ladle/pour about 1 cup of the pasta water in the glass measuring cup. Drain the pasta in the colander and let it rest, covered loosely with a pot lid or plate.
Place a small, heat-safe bowl near the stove. In a large cast-iron or non-stick skillet, heat 3 tablespoons of olive oil over medium-low heat. When the oil is hot, add the red pepper flakes and garlic and cook, stirring constantly, until the garlic begins to simmer. Cook for about 30 seconds more to infuse the oil with spicy, garlicky flavor, but do not let the garlic brown. Pour and scrape the seasoned oil into the heatproof bowl and set aside. Wipe out the pan with a clean kitchen rag or paper towel.
Return the pan to the stove. Add 2 tablespoons olive oil and heat over medium-low until shimmering. Add the broccoli and sprinkle with 1 teaspoon salt. Cook, stirring occasionally, until the broccoli has shrunk to a single layer in the pan and turned bright green, and most have some browning on them (about ten minutes). Don't quit cooking prematurely here; you want the broccoli to be nice and toasty.
Get out the pan's lid or a cookie sheet and keep it handy. Add the sun-dried tomatoes to the pan. Measure out 1/3 cup pasta water (keep the rest for later) and pour it into the pan. Cover the pan with your lid or cookie sheet and continue cooking until the water has simmered down to almost nothing, about 15 to 30 seconds. Uncover and remove the pan from heat.
Add the drained pasta to the pan and drizzle in all of the infused oil. Give it a stir, then add the goat cheese and most of the Parmigiano. Stir until everything is well distributed. Add another 1 to 2 tablespoons pasta water, the chopped olives and lemon juice, and stir until the goat cheese loosens up and gets creamier. Season to taste with salt and add a tablespoon more pasta water or additional goat cheese if you'd like it to be more creamy. If it seems dry at all, add a little splash of olive oil and mix well. Mix in the chickpeas and/or arugula, if using.
Serve right away, garnished with the remaining Parmigiano.