Start by cutting the broccoli up into very small florets, about 1/2″ across. If you buy a bag of florets, cut them smaller if needed, they often come in very large pieces.
Place a large pot over medium-high heat and sauté the chicken in 1 tbsp of the butter. When the chicken is almost done, but not quite, remove it from the pan and set it aside.
Add the pasta, water, part of the heavy cream, and salt. Stir to combine while bringing the mixture to a boil. Cover and cook stirring occasionally until the pasta is barely softening, about 2-3 minutes.
Add the broccoli florets to the pasta, cover, and continue to cook over a simmer, stirring occasionally until the broccoli and pasta are tender. Add more water if necessary if there is too much evaporation.
When the pasta is al dente, add the chicken back to the pan to finish cooking. Add the remaining cream and part of the parmesan, and season to taste with sea salt and pepper.