01 - Melt the butter in a large pot over medium heat. Add the onion and carrot, sautéing until softened, about 5 minutes.
02 - Stir in the garlic and cook for 1 minute until fragrant.
03 - Sprinkle in the flour, stirring constantly for 1-2 minutes to form a roux.
04 - Gradually whisk in the vegetable broth, ensuring no lumps remain.
05 - Add the chopped broccoli, dried thyme, salt, and pepper. Bring to a simmer, cover, and cook for 15 minutes until the broccoli is tender.
06 - Reduce the heat to low. Stir in the milk and cream, and simmer gently for 5 minutes without boiling.
07 - Using an immersion blender (or working in batches with a countertop blender), blend the soup until smooth or leave some chunks for texture, as desired.
08 - Stir in the grated cheddar cheese, mixing until melted and the soup is creamy. Taste and adjust seasoning if needed.
09 - While the soup finishes, toast the bread slices until golden.
10 - Ladle the hot soup into bowls and serve with toasted bread on the side.