Preheat the oven to 425 degrees F. Place the broccoli on a baking sheet and drizzle with the olive oil. Sprinkle with salt and pepper. Roast the broccoli for 15 to 20 minutes, until it just starts to char.
Bring a pot of salted water to a boil. Once boiling, cook the pasta according to the directions on the package. Remember! You want to reserve at least 1 cup of the pasta water once it has finished cooking.
Heat a large skillet over low heat. Add the butter and the garlic. Cook, stirring often, for 5 to 6 minutes until the garlic slowly begins to brown.
In a bowl, whisk together the eggs, parmesan cheese and almost all of the lemon zest (I reserve a bit for sprinkling). Add a few cracks of black pepper.
Once the pasta is cooked, drain it well. Toss it in the garlic and butter. Turn the heat off under the skillet. Add the parmesan/egg mixture and immediately begin stirring with a spatula to create a creamy sauce. Stir for at least 2 minutes, so the eggs don’t scramble and a sauce coats the noodles. Whisk in ½ cup of the pasta water and stir until the sauce comes together. If you want the sauce even more liquidy, you can add the remaining pasta water!
Fold the roasted broccoli into the pasta in the skillet. Sprinkle with more parmesan and black pepper. Serve immediately!