Bring a large pot of water to a bowl. Season the water with salt.
While waiting for the water to boil, heat a cast iron skillet (or any large skillet) over medium high heat with 1 tablespoon olive oil. Add the broccoli, season with a pinch of salt and pepper, and cook the broccoli uncovered for 4 minutes until bright green. Add in 1/4 cup water, and cook uncovered until the water has evaporated, and the broccoli is crisp tender. Transfer the cooked broccoli to a medium size bowl.
Add the fettuccine to the boiling water, and cook according to manufacturers instructions.
In the same skillet you cooked the broccoli, heat another tablespoon of olive oil, and cook the chicken in an even layer. Season with salt and pepper. Cook about 5-7 minutes, or until the chicken is golden brown and cooked thru, turning the chicken several times. Transfer the cooked chicken to the same bowl as the broccoli.
In the same skillet, melt the butter, then cook the garlic for 30 seconds, or until fragrant. Then add the heavy cream, and scrape up any brown bits from the bottom of the skillet. Add the grated parmesan cheese, and cook until melted and just starting to bubble. Turn the heat to low.
When the pasta is done cooking, reserve 1 cup of the pasta cooking water. Drain the pasta from the water, then add the cooked pasta to the skillet with the sauce, and toss to coat the noodles. Add in the cooked chicken and broccoli, toss to combine. If necessary, add in pasta water to thin the sauce to desired consistency.