Place a large saucepan of generously salted water over high heat and bring to a boil.
Meanwhile, in another large saucepan over medium-low heat, add enough olive oil to coat the bottom of the pan. Add the garlic and cook until soft and translucent but without letting it brown, 5 to 8 minutes. Remove from the heat.
Add the pasta to the boiling water and cook according to the package directions.
When you have just a few minutes cooking time for the pasta left, set the pan containing the garlic back over medium-high heat, add the broccoli, and stir-fry until the broccoli is crisp and green and tender, about 3 minutes, depending on the size of the florets.
Using tongs, lift the pasta from the water and allow any excess water to drip back into the pan. Add the pasta to the broccoli, along with a few spoonfuls of the pasta cooking water. Gently toss to combine and then add a good drizzle of olive oil.
Add the olives, chile flakes, and lemon zest and toss to combine. Stir in the feta and toss to evenly distribute the ingredients.
Taste and, if desired, season with salt and black pepper. If the dish seems a little dry, stir in a little more pasta water, about 1 tablespoon at a time, using as much as 1/2 cup to create a sauce if desired.Serve immediately.