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.
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.
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.