Melt butter in a large dutch oven or pot over medium-high heat. Add onion and cook for 2-3 minutes or until softened and light gold. Add the garlic and saute for another minute.
Add flour and whisk for 1-2 minutes or until the flour begins to turn golden brown. Pour in the stock then add the broccoli florets, carrots, and seasonings. Bring the soup to a boil then reduce heat to medium-low heat and simmer for 15-20 minutes or until the broccoli and carrots are cooked through.
Stir in half & half and cheddar cheese and simmer for another minute or so. Taste and add salt and pepper, as needed. Sprinkle with red pepper flakes, if desired.
Serve with crackers, biscuits, toasted crusty bread, or in a bread bowl if desired.