Sauté the Aromatics:
Heat butter or olive oil in a large pot over medium heat. Add the diced onion and sauté for 3-4 minutes until softened. Add the minced garlic and cook for 1 minute, stirring frequently.
Cook the Vegetables:
Add the diced potatoes, broccoli florets, broth, thyme, salt, and pepper to the pot. Bring the mixture to a boil, then reduce the heat and simmer for 20 minutes, or until the potatoes and broccoli are tender.
Mash or Blend:
Use a potato masher to partially mash the soup for a chunkier texture, or use an immersion blender to purée the soup until smooth if you prefer a creamier consistency.
Add Dairy and Cheese:
Stir in the milk or cream, followed by the shredded cheddar cheese. Mix until the cheese is fully melted and the soup is creamy.
Taste and Adjust:
Taste the soup and adjust seasoning with more salt or pepper if needed.
Serve:
Ladle the soup into bowls and garnish with extra shredded cheese, croutons, or parsley if desired.