Melt the butter in a large pot over medium heat.
Add the diced onion and carrots and cook until softened, about 5 minutes.
Stir in the minced garlic and cook for 30 seconds.
Sprinkle the flour over the vegetables and stir until fully coated.
Cook the flour for 1 minute to remove the raw taste.
Slowly pour in the chicken broth while whisking to dissolve the flour.
Add the half-and-half and stir until smooth.
Add the broccoli florets and bring the soup to a gentle simmer.
Reduce heat to medium-low and cook for 10 to 12 minutes, or until the broccoli is tender.
Stir in the salt, pepper, paprika, and garlic powder.
Reduce heat to low.
Slowly add the shredded cheddar cheese, stirring continuously until melted and smooth.
Taste and adjust seasoning if needed.
Simmer for 2 more minutes, then remove from heat.