Cook spaghetti until al dente according to package instructions, reserving 1 1/2 cups of the cooking water.
When pasta has about 5 minutes left to cook, heat 2 tablespoons of the olive oil in a large, non-stick skillet over medium heat. Add garlic, broccoli and kosher salt; cook and stir until broccoli is tender and garlic is lightly browned, about 5 minutes (see recipe note #1 for softer broccoli). Add cooked pasta, remaining 2 tablespoons of olive oil, red pepper flakes and 1/2 cup of the reserved pasta water (plus more as desired); cook and stir until heated through, 2 to 3 minutes. Season to taste with salt and pepper. Transfer to serving bowl and sprinkle parmesan cheese evenly over the top; serve.
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Notes
At this point the broccoli will be crisp-tender. If you prefer a softer broccoli, add 2 tablespoons of the pasta cooking water and cover to steam to your desired tenderness (2 to 4 minutes or longer), adding more water if needed. Then add cooked pasta and proceed with recipe.
If you have leftover pasta, save any remaining cooking water to add to the pan while reheating.