Fill a large pot with water (about 3 quarts) and bring to a boil. Season generously with salt (about 1 tbsp). Add the broccoli to the boiling water and let cook for 1 minute, or until no longer frozen (if using fresh broccoli, cook until very tender, maybe 2 minutes). Use a fine mesh sieve to remove the broccoli and transfer to a high-speed blender.
Add the pasta to the boiling water and cook according to the package instructions. Reserve at least 1 cup of pasta water, then drain and transfer to the pot.
To the blender, add the basil, spinach, olives, butter, lemon, garlic and salt. Add 1 cup of the reserved pasta water and blend until smooth, using a tamper tool to encourage everything to blend.
Pour the sauce into the pot with the pasta and stir to coat the noodles. Serve topped with marinated tuna (if using), cheese, pepper, and walnuts.