Heat the oil in a large skillet over medium-high heat.
Skip the next 3 steps if you are using already-cooked chicken.
Add the chicken and cook without stirring until browned, 2-3 minutes.
Stir and continue to cook 1 minute longer. (It won't be fully cooked.)
Transfer to a bowl and set aside.
Put the garlic, pepper flakes, and scallions into the skillet and cook until fragrant, about 30 seconds.
Add the pasta, 1-1/4 cups of the milk, and the broth.
Bring to a boil over high heat and cook, stirring frequently, until it forms a thick sauce, about 12 minutes.
Stir in the broccoli, sun-dried tomatoes, and the remaining milk.
Cover, reduce heat to medium, and cook until the broccoli turns bright green and is almost tender, 3-5 minutes.
Stir in the cheese and chicken.
Simmer, uncovered until the chicken is cooked through, 1-2 minutes.
Season with pepper and salt (if needed).