Melt the butter in a large pot or deep skillet over medium heat. Add the minced garlic and sauté for about 1 minute until fragrant.
Pour in the broth and milk, then add the salt, pepper, mustard powder, and garlic powder. Bring to a gentle boil.
Add the uncooked pasta and stir well. Return to a boil, then reduce heat to medium and cook for 8 minutes, stirring occasionally.
Add the broccoli florets and continue cooking for another 4-5 minutes until both the pasta and broccoli are tender.
Remove from heat and stir in the shredded cheddar cheese, mixing until completely melted and creamy.
Let the pasta rest for 2-3 minutes to thicken up before serving. The sauce will continue to thicken as it stands.
Taste and adjust seasonings if needed. Serve hot with extra shredded cheese on top if desired.