Cook your broccoli florets in a large stock pot in salted water for 5 minutes.

Drain and set aside.

In the pot, melt 1/4 cup of butter and saute the onion and carrots, stirring until soft about 3 minutes.

Remove the carrots and onion from the pan, place in a bowl and set aside.

Melt the remaining 1/2 cup of butter in the pot on a medium/low heat.

Add the flour and mix together until incorporated.

Slowly add the half and half, and the milk, while whisking together.

When it becomes slightly thickened, add the chicken stock and whisk to combine.

Turn the heat to low, cover, and simmer for 15 to 20 minutes.

Add the broccoli, carrots and onions. Simmer for another 5 minutes.

Stir in 3 cups of cheddar cheese until melted.

Season with salt and pepper to taste.

Serve hot, garnished with remaining cheddar cheese.