Melt butter in a large pot over medium-high heat; add the onion and cook 3-4 minutes until softened; add the garlic and sauté for another minute
Add flour; whisk for 1-2 minutes until the flour begins to turn golden in color; pour in the chicken stock, broccoli florets, carrots and seasoning; bring to a boil then reduce heat to medium-low; simmer for 15 minutes until the broccoli and carrots are cooked through
Stir in half & half and cheddar cheese; let simmer for another minute; taste and adjust seasoning if needed