In a large pot over medium heat, drizzle some olive oil and add the chopped onion. Sauté for about 5 minutes until translucent and fragrant. Stir in minced garlic and cook for an additional minute.

Add the diced potatoes and chopped broccoli to the pot, stir to coat in oil, and sauté for another 3-4 minutes.

Pour in vegetable or chicken broth and bring to a vigorous boil. Reduce heat to low and let it simmer for about 15-20 minutes until the potatoes are fork-tender.

Using an immersion blender, puree the soup until smooth, or transfer to a regular blender in batches to blend until velvety.

Return the pot to low heat, stir in milk or cream and shredded cheddar cheese until melted and creamy.

Season with salt and pepper to taste, adding optional nutmeg or hot sauce if desired.