Preheat oven to 375°F. In a 10-inch cast iron skillet (or other oven-proof skillet), melt butter over medium-high heat. Add carrot and onion; cook, stirring occasionally, until slightly softened, about 3 minutes. Whisk in the flour; cook, stirring constantly, for 2 minutes. Gradually whisk in the broth and milk until smooth. Bring to a boil, reduce the heat, and simmer, whisking frequently, until thickened, about 7-10 minutes. Remove from the heat; stir in the broccoli, 1 cup of grated cheddar, salt, thyme, and black pepper. Taste and season with additional salt and pepper, if desired.