Place the onion and garlic into the 40-ounce processor bowl. Pulse 3–5 times until a small chop is achieved.
Heat canola oil in a 3-qt saucepan over medium heat and sweat out the onions and garlic for 5 minutes until translucent. Add the broccoli florets to the pot and cook 1–2 minutes. Add vegetable broth and bring to a boil, then reduce heat to low and cook 25 minutes until broccoli is tender. Add half & half and cheese. Remove from heat and allow to cool to room temperature.
Working in two batches, place half of the cooled soup into the processor bowl and blend until smooth.
Return soup to saucepan and simmer until heated through. Season with salt and pepper to taste.
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