Place a heavy bottomed pot over medium-high heat. Add 2 to 3 tablespoons of olive oil and then toss in the onion. Season with salt and pepper and cook for 3 to 5 minutes, or until just translucent. Add the broccoli and again season with salt and pepper.
Pour the chicken stock over the broccoli and bring up to a boil and then reduce to a simmer. Cook for 8 to 10 minutes, or until the broccoli is fork tender.
Let cool slightly and then transfer, working in batches, to a blender. Cover the blender with a towel to ensure it doesn't splatter, and puree until smooth. Taste and adjust seasoning.
Place another heavy bottomed pot over medium high heat and add 1 to 2 tablespoons of olive oil. Add the sausage and brown on all sides, about 8 to 10 minutes.
When almost completely browned, add the kale. Season with salt and pepper and cook for 3 to 4 minutes. Add the beans and cook for another 1 to 2 minutes.
Pour the broccoli soup in the sausage and kale and stir to combine. Let cook for one to two more minutes to let the flavors meld. Taste and adjust the seasoning, then serve while hot.