Prepare the vegetables. In a big pot, warm the olive oil on medium heat. Put the onion and garlic in the pan and cook until they are soft and smell good, which should take around 3 minutes.
Cook the soup slowly. Put the chopped potatoes, broccoli, carrots, broth (chicken or vegetable), and thyme in the pot. Add salt and black pepper. Heat until boiling, then lower the heat and let it cook gently for 15–20 minutes, or until the potatoes and vegetables are soft.
Mix the soup. Use a hand blender to blend the soup partially, so it still has some pieces for texture. You can also put half of the soup in a blender, blend it until it's smooth, and then pour it back into the pot.
Include cream and cheese. Add the thick cream and grated cheddar cheese. Mix well until the cheese is completely melted and the soup becomes creamy. Add salt and pepper to taste, if necessary.
Wait on tables. Put the soup in bowls, add some chopped chives on top, and savor it!