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Step 1
Melt 1 Tbsp butter in a pot over medium heat.
Step 2
Add onion and saute until tender, about 6 to 8 minutes. Add garlic and saute 30 seconds longer.
Step 3
Pour in broth and broccoli, and season with salt and pepper to taste.
Step 4
Bring to a simmer, cover, then reduce heat to medium-low. Simmer until broccoli is tender about 6 to 10 minutes.
Step 5
Meanwhile in a medium saucepan set over medium heat melt remaining 5 Tbsp butter. Add flour and cook, stirring constantly, 1 - 2 minutes.
Step 6
Slowly pour half and half into flour mixture while whisking. Season with salt to taste. Cook mixture, whisking constantly, until thickened (nearly simmering). Remove from heat and stir in parmesan.
Step 7
Once broccoli is tender remove soup from heat and add ice to slow it from cooking further. Using an immersion blender process mixture until broccoli is pureed into soup.
Step 8
Pour white sauce mixture into soup and stir to combine. Serve soup warm.