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Cook pasta according to package instructions. Reserve ½ cup of cooking water. Drain and set aside.
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In a dry pan over medium heat, toast walnuts for a few minutes, until golden brown, Remove from heat.
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Place ⅓ cup walnuts, broccoli, basil, garlic, vegetable broth, lemon juice, and crushed red pepper flakes to a blender. Start blending and slowly stream in olive oil. Add more olive oil if needed. Taste and season with salt.
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Toss the pesto with the hot pasta. Add small amounts of the pasta water if it seems to thick. Sprinkle with the remaining walnuts and extra Parmesan cheese, if desired.