Trim the end of the broccoli stem and separate the florets from the stalk. Cut the stalk into smaller pieces.
Bring about 5 liters of water to a boil in a large pot and add a tablespoon of rock salt. Cook the broccoli for about 5 minutes until tender but still crunchy.
In a medium frying pan over low heat, add extra virgin olive oil. Crush the garlic cloves and add them to the oil, letting them simmer until lightly browned. Add 1 to 2 tablespoons of water to prevent burning.
Transfer the cooked broccoli to the pan with garlic and oil. Crush some of the broccoli with the back of a spoon for texture.
Return the pot of water to a boil. Add spaghetti and cook according to package instructions until al dente. Drain the pasta and add it to the pan with the broccoli mixture.
Add salt and pepper to taste, then pour in about half a cup of reserved pasta water and mix well.
Stir in grated pecorino cheese until melted and well combined.
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Notes
Store leftovers in an airtight container in the refrigerator for up to 2 days. Reheat gently on the stove or in the microwave.