In a large pot, melt 4 tablespoons of butter over medium-high heat until it bubbles. Add 1 chopped yellow onion and sauté for 4-5 minutes until it turns translucent and soft.

Stir in 3 minced garlic cloves and 1/4 cup of all-purpose flour, cooking for about 1 minute until well blended and bubbly.

Gradually whisk in 2 cups of low-fat milk and 2 cups of low-sodium chicken broth until smooth. Bring the heat to high until it reaches a gentle simmer.

Add in 4 cups of fresh broccoli florets. Reduce the heat to medium and cook for 6-8 minutes until the broccoli is tender.

Remove from heat and blend part of the mixture for a smoother texture, keeping some broccoli chunks.

Return to low heat and stir in 1 cup of heavy cream, 2 cups of sharp cheddar cheese, and 1/2 cup of grated Parmesan cheese until melted.

Season with salt and pepper to taste, then ladle into bowls and garnish with extra cheese if desired.