01 - Melt the unsalted butter in a large pot over medium heat. Add diced onion and carrot, sautéing for approximately 4 minutes until softened.
02 - Incorporate minced garlic and cook for 1 minute until fragrant.
03 - Sprinkle all-purpose flour over the vegetables, stirring constantly for 2 minutes to create a roux.
04 - Gradually whisk in the vegetable broth, ensuring a smooth mixture without lumps.
05 - Add chopped broccoli to the pot. Bring to a simmer and cook uncovered for 10 to 12 minutes until the broccoli is tender.
06 - Lower heat to low. Puree the soup with an immersion blender until smooth, or leave some pieces of broccoli for texture as desired.
07 - Stir in whole milk and heavy cream. Warm the mixture gently, avoiding boiling.
08 - Gradually add grated sharp cheddar, stirring until fully melted and smooth. Season with salt, black pepper, and an optional pinch of ground nutmeg.
09 - Taste and adjust seasoning as necessary. Serve hot, optionally garnished with extra cheddar or fresh herbs.