In a large pot, heat a splash of oil over medium heat. Add the diced onion and carrot, sautéing until softened, about 5 minutes.

Add the minced garlic and broccoli florets, cooking for an additional 2-3 minutes.

Pour in the vegetable broth and bring to a boil. Reduce heat and let simmer for 15 minutes until broccoli is tender.

Using a blender, carefully puree the soup until smooth, or leave it slightly chunky if preferred.

Return the pureed soup to the pot over low heat. Stir in the milk and shredded cheddar cheese until melted and combined.

Add the cooked chicken, salt, pepper, and thyme, stirring until heated through.

Ladle the soup into bowls and serve hot, garnished with additional cheddar cheese if desired.