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