Melt the butter in a large Dutch oven or heavy-bottomed pot over medium heat.
Add the diced onion and sauté for 3 to 4 minutes until translucent and softened.
Whisk in the flour and cook for 1 to 2 minutes to form a roux, ensuring it does not brown.
Gradually whisk in the chicken stock and half-and-half, stirring constantly to ensure a smooth texture without lumps.
Allow the liquid to simmer for 10 minutes over medium-low heat until thickened.
Incorporate the broccoli florets and shredded carrots into the pot.
Cover and simmer for an additional 10 to 12 minutes, or until the broccoli reaches desired tenderness.
Stir in the ground nutmeg, salt, and black pepper.
Remove the pot from the heat and stir in the shredded cheddar cheese one handful at a time until completely melted and the soup is homogeneous.
Adjust seasoning to taste and serve immediately.