Sauté the onions: In a spacious saucepan, heat the butter on medium heat until it melts. Introduce the diced onion and cook, stirring occasionally, until it becomes tender and translucent, which should take around 3 to 4 minutes.

Make the roux: Stir in the flour and cook for 1-2 minutes, stirring constantly, until the flour is lightly browned and forms a paste. This will help thicken the soup.

Add liquids: Gradually incorporate the chicken broth and half-and-half cream by stirring continuously. Keep stirring to avoid lumps. Bring the mixture to a simmer, allowing it to thicken slightly.

Cook the vegetables: Incorporate the broccoli florets and shredded carrots into the saucepan. Simmer for about 15-20 minutes, or until the vegetables are tender but not mushy.

Season and blend: Stir in the grated cheddar cheese, a sprinkle of nutmeg, and season with salt and pepper to taste. You can blend part of the soup for a smoother texture, leaving some chunks of broccoli for added bite.

Serve: Ladle the hot soup into bread bowls or regular bowls, and garnish with extra cheese if desired. Enjoy the warm, cheesy goodness!