In a soup pot, over medium heat, melt butter. Add onions and garlic, adjusting heat lower as needed, and saute until translucent and very soft, about 8-10 minutes. Whisk in flour and cook 6-8 minutes or until golden.
Remove pot from heat and slowly add chicken broth, whisking until flour mixture is dissolved and smooth. Add half and half, Better than Bouillon, Bay Leaves, salt, pepper, nutmeg, and bring to a simmer. Reduce the heat to medium-low and cook, uncovered, until slightly thickened, about 20 minutes.
Add the carrot and broccoli and simmer, uncovered, until veggies are soft, about 20 minutes. Remove bay leaves and discard. Ladle soup into a blender, in batches, and puree until smooth (flecks of the veggies are okay). Return to the pot. Add heavy cream and gently heat through. Adjust seasoning and serve.