Sauté the Aromatics: In a large pot, melt butter over medium heat. Add the diced onion and cook until soft, about 3 minutes. Stir in the garlic and cook for another 30 seconds until fragrant.
Add the Vegetables: Pour in the broth, then add the diced potatoes and broccoli. Bring to a boil, then reduce heat and let simmer for 15 minutes, or until the potatoes are fork-tender.
Make the Cheese Sauce: In a small bowl, whisk the flour with a few tablespoons of milk to form a smooth paste. Pour the remaining milk into the soup, then slowly stir in the flour mixture. Let the soup simmer for another 5 minutes to thicken.
Add the Cheese: Remove the pot from heat and stir in the shredded cheddar cheese until fully melted and smooth. Season with salt, pepper, and paprika if using.
Blend (Optional): If you prefer a smoother texture, use an immersion blender to puree part of the soup while leaving some chunks.
Serve and Enjoy: Ladle the soup into bowls and garnish with extra cheddar cheese or a sprinkle of black pepper. Serve warm with crusty bread or crackers.
