Melt butter in a large skillet or deep sauté pan over medium heat. Add minced garlic and cook for 30 seconds until fragrant, stirring constantly to prevent browning.
Add dry orzo pasta to the skillet and stir to coat with butter. Toast for 1-2 minutes, stirring frequently, until the orzo is lightly golden and nutty.
Pour in chicken broth and heavy cream. Add salt, pepper, and paprika if using. Stir well to combine and bring to a gentle simmer over medium heat.
Once simmering, reduce heat to medium-low. Cook uncovered for 8 minutes, stirring occasionally, until the liquid begins to thicken and the orzo is mostly cooked through.
Add broccoli florets to the skillet. Continue cooking for 4-5 minutes, stirring frequently, until the orzo is al dente and the broccoli is tender-crisp. Most of the liquid should be absorbed into a creamy sauce.
Remove from heat. Gradually stir in freshly grated cheddar cheese, one handful at a time, until completely melted and the sauce is smooth and glossy.
Taste and adjust seasoning with additional salt and pepper if needed. If the sauce is too thick, thin with a splash of warm milk or broth.
Serve immediately while hot and creamy. Garnish with fresh chopped parsley and extra cheese if desired.