Heat oil in a deep 12-inch skillet or wide pot over medium-high heat. Cook chicken and onion for 5 minutes, stirring, until the chicken is opaque outside; add garlic for 30 seconds.
Add dry pasta, broth, salt, paprika and pepper. Bring to a simmer, cover and cook over medium-low heat for 6 minutes, stirring once from the bottom.
Add broccoli and stir gently. Cover and cook for 5 to 7 minutes more, stirring twice, until the pasta is tender and the thickest chicken pieces reach 165°F.
Add reserved water in 2-tablespoon amounts only if the pot starts drying before the pasta is tender. Uncover and simmer for 1 to 2 minutes if too much liquid remains.
Remove from heat and stir in cheddar until smooth. Rest for 2 minutes, stir once more and serve with the broccoli evenly distributed.