Bring a large pot of salted water to a boil over high heat. Fill a large bowl with ice water. Cook the broccoli in the boiling water for about 5 minutes, or until it is completely tender. Using a slotted spoon, transfer the broccoli to the ice water and cool completely. Drain, pat dry, and coarsely chop the broccoli .
Add the penne to the same pot of boiling water and cook, stirring often, for about 12 minutes, or until it is tender but still firm to the bite. Drain, reserving 4 tablespoons of the pasta water. Return the drained pasta to the dry pot and set aside.
Meanwhile, heat a large heavy frying pan over medium-high heat. Add 2 tbsp. of the oil, then add the garlic, chilli, and anchovy. Cook for about 1 minute, or until the garlic is toasted. Add the broccoli and cook, stirring, for about 3 minutes, or until it is warm and beginning to release its liquid. Stir in 1/2 tablespoon of the lemon juice.
To the pasta, add the broccoli mixture, 3 tbsp. of the cheese, and the remaining tablespoon oil. Toss to combine and season to taste with salt and more lemon juice. Add enough of the reserved pasta water to moisten as desired. Divide among 2 bowls, garnish with more cheese, and serve.