Melt 2 tablespoons of the butter with the olive oil in a large stockpot over medium heat. Add the chopped onions and carrots and sauté until very tender, about 10 to 12 minutes. Stir in the chopped garlic and sauté for an additional 3 minutes.
Toss in the remaining 2 tablespoons of butter. When the butter has melted add the flour, stirring to blend into the vegetables then add the kosher salt and pepper.
Pour in the chicken stock, stirring until smooth and well blended.
Add the chicken base, nutmeg, dry mustard, lemon zest and the bay leaf. Simmer for about 10 minutes to thicken slightly.
Pour in 1 cup of the half and half plus the milk. Add the broccoli, nestling into the soup base. Reduce the heat to medium-low, cover the pan and cook for 30 minutes or until the broccoli is tender, stirring occasionally.
Reduce the heat to low, pour in the remaining cup of half and half and blend in the cheese. Stir until the cheese has completely melted into the soup.
To serve, top with searing prosciutto or crispy bacon and a touch of extra cheese.