Melt butter to a large heavy-bottomed pot over medium heat.
Add onions and cook until lightly golden brown on the edges and the onions are translucent about 4-5 minutes.
Make a roux by stirring in the flour. Continue to stir and cook the flour mixture until the roux turns the color of peanut butter. This will take about 4 minutes.
Grab a whisk and slowly pour in the chicken broth and half and half while whisking constantly. Turn the heat up to medium-high and bring the soup to a slow bubble. Once bubbling, reduce the heat to simmer.
Stir in the broccoli florets and carrots. Simmer the soup for 15 minutes, stirring every 3-4 minutes.
Stir in the salt and cheddar cheese until the cheese is nice and melted. Season the soup to taste with additional salt. Serve.