Stir melted butter and flour over low to medium heat for about 3 minutes, and then slowly add the milk and chicken broth, while stirring.
Cook for about 15 minutes on low to medium heat. While that’s cooking, I like to use my potato peeler to shave up the carrot, and then use a knife to chop the slices into bite sizes. My hubby’s not a huge fan of cooked carrots so this makes them nice and thin, and small. I love the colorful addition.
Then add the broccoli florets and carrot slices to the broth mixture. Simmer for 15 minutes. Check broccoli florets to see if they’re tender, and if they’re not, cook for 5 more minutes.
Remove from heat. If the florets are bigger than I like, I VERY CAREFULLY use my kitchen scissors to cut the broccoli up into smaller bite size pieces.
Then stir in the cheddar cheese till it’s melted and incorporated, and you're ready to serve.