Sauté Onions and Garlic: In a large pot, melt the butter over medium heat. Add the onion and garlic, cooking until translucent and fragrant.
Add Broccoli and Carrots: Introduce the broccoli florets and julienned carrots to the pot, stirring to mix with the onions.
Pour in Broth: Add the chicken or vegetable broth. Bring the mixture to a boil, then reduce to a simmer until the broccoli is tender, about 10-15 minutes.
Thicken with Flour: Sprinkle the flour over the vegetables, stir to coat, and cook for another minute. Slowly add the milk while continuously stirring to prevent lumps.
Cook Until Thickened: Continue to cook on medium heat until the soup thickens.
Blend (Optional): For a smoother texture, blend the soup using an immersion blender or transfer to a blender in batches.
Add Cheese: Remove the pot from the heat. Gradually stir in the shredded cheese until melted and smooth. Season with salt, pepper, and optional paprika.
Serve Warm: Adjust seasonings if needed and serve the soup warm with crusty bread or a fresh salad.