1. In a large pot, melt the butter over medium heat. Add the chopped onion and sauté until it becomes translucent, about 5 minutes. Stir in the garlic and cook for an additional minute, enjoying the fragrant aroma that fills your kitchen.
2. Add the broccoli florets and diced potatoes to the pot. Pour in the chicken or vegetable broth, bringing the mixture to a simmer. Cover and cook for about 15 minutes, or until the potatoes are tender.
3. Using an immersion blender, carefully blend the soup until it reaches your desired consistency. If you prefer a chunkier texture, blend only half of the soup.
4. Reduce the heat to low and stir in the milk and shredded cheddar cheese. Stir continuously until the cheese has melted completely. Add salt, pepper, and paprika to taste. If using, pour in the heavy cream for a richer finish.
5. Allow the soup to heat through for another 5 minutes, stirring occasionally. Taste and adjust seasoning as needed.

Remember, cooking is an art, not a science. Feel free to adjust the cheese or spice levels to suit your palate, much like my grandmother taught me to follow my senses rather than measurements.