Bring a large saucepan with plenty of water for the penne and broccoli florets (they are cooked together) to boil.
Roast the pine nuts in a pan without oil. Put the seeds in a bowl. Use the same pan to briefly sear the garlic in a little olive oil.
Cut the bokkoli florets into bite-sized pieces and set aside. Divide the broccoli stalk into pieces and puree together with the water in the chopper until creamy. Pour into a bowl. Then puree the fried garlic, 2/3 of the pine nuts (save the rest for garnish), basil, olive oil, salt, lemon zest and lemon juice in the food processor. Mix with the pureed broccoli stalk.
Put the penne in the boiling and salted water and cook until al dente. 5 minutes before the end of the cooking time, add the broccoli florets to the boiling water and let them cook as well. Drain the penne and broccoli, collecting some of the cooking water.
Return the penne and broccoli to the pot and mix with the pesto. Depending on the desired consistency, add some of the cooking water. Fill onto plates and serve garnished with the remaining pine nuts.