Bring a large pot of salted water to a boil. Add the pasta and cook according to the package directions. During the last 2 minutes of cooking, add the broccoli.
While the pasta cooks, add the pancetta to a large skillet over medium heat and cook until crispy, 5 to 6 minutes. Spoon off all but 1 tablespoon of the fat and discard. Add the garlic to the pan and sauté until it softens slightly, about 1 minute.
In the meantime, whisk together the Parmesan, eggs and some salt and pepper in a bowl, then set aside.
When the pasta and broccoli are cooked, reserve 1 cup pasta cooking water, then drain. Add the pasta and broccoli to the skillet along with 1/4 cup of the pasta water and toss.
While whisking, whisk 1/4 cup of the pasta water into the egg mixture.
Turn the heat off under the skillet and pour the egg mixture over the pasta, tossing quickly so the eggs don't scramble. Add a little more of the pasta water to thin out the sauce, if necessary. Serve immediately with lots of black pepper and a sprinkling of grated Parmesan.