Melt 1 tablespoon of the butter in a medium saucepan over medium heat, then sauté the onion for 5-7 minutes until lightly caramelized. Transfer to a bowl and set aside.
1/4 cup + 1 tablespoon salted butter, 1/2 medium onion
Melt the remaining butter in the same pan, then whisk in the flour. Cook for 2-3 minutes until thickened to create a roux.
1/4 cup all-purpose flour
Slowly whisk in the half-and-half or whole milk. The mixture will thicken quickly. Whisk in the chicken broth, then bring to a simmer.
2 cups half-and-half or whole milk, 2 cups chicken broth
Add the broccoli, grated carrots, and reserved onions, then season with salt and pepper. Cover and simmer 15-20 minutes until the vegetables are tender.
1/2 pound fresh broccoli, 1 cup grated carrot, 1 teaspoon kosher salt, 1/4 teaspoon freshly ground black pepper
Remove 1-2 cups of the vegetable chunks and transfer to a blender. Cover the blender with a towel and blend to break up the chunks. Return to the pot of soup and stir to combine.
Reduce heat to low, then stir in the grated cheddar cheese until melted. Remove from heat and serve immediately.
8 ounces freshly grated sharp cheddar cheese