01 - Melt butter in a large pot over medium heat. Add chopped onion and diced carrot, sautéing until softened and fragrant, approximately 5 minutes.
02 - Stir minced garlic into the pot and cook for 1 minute until fragrant, being careful not to brown the garlic.
03 - Sprinkle flour evenly over the vegetables, stirring constantly for 1-2 minutes to cook out the raw flour taste and create a smooth roux base.
04 - Gradually whisk in vegetable broth, ensuring the mixture remains smooth and free of lumps. Continue whisking until fully incorporated.
05 - Add broccoli florets, dried thyme, salt, and pepper. Bring to a gentle simmer, cover, and cook for 15 minutes until broccoli is completely tender.
06 - Reduce heat to low. Stir in whole milk and heavy cream, allowing the soup to simmer gently for 5 minutes. Avoid boiling to prevent dairy separation.
07 - Use an immersion blender to puree the soup until completely smooth, or blend in batches using a countertop blender. For added texture, leave small chunks of broccoli intact.
08 - Add grated sharp cheddar cheese gradually, stirring continuously until fully melted and incorporated. Taste and adjust seasoning with additional salt and pepper if needed.
09 - While soup finishes cooking, toast bread slices in a toaster or oven until golden brown and crisp on the exterior.
10 - Ladle hot soup into serving bowls and accompany with warm toasted bread on the side for dipping.