In a large pot, melt 1 tablespoon of unsalted butter over medium heat. Add the chopped onion and sauté for about 5 minutes until it becomes translucent and fragrant.
In a separate bowl, combine 1/4 cup melted unsalted butter and 1/4 cup flour. Whisk until smooth. This roux will thicken your soup.
Gradually whisk in the 3 cups of half-and-half and 3 cups of chicken stock into the pot with the sautéed onion. Continue to stir until well combined.
Stir in the fresh broccoli florets and shredded carrots. Allow the mixture to come to a gentle simmer, then reduce the heat to low.
Add the nutmeg, salt, and pepper to taste. Then, gradually stir in the roux mixture. Allow the soup to cook for an additional 10-15 minutes, stirring occasionally, until the vegetables are tender.
Remove the pot from the heat and stir in the grated sharp cheddar cheese until melted and incorporated. Taste and adjust the seasoning as needed.
Ladle the Better Than Panera Broccoli Cheddar Soup into bowls and serve hot. You can top it with extra cheese or fresh herbs if desired!