Into a small saucepan filled, add cashews and just enough water to cover by 1 inch. Bring to a boil and allow cashews to boil for 8 minutes. Drain cashews.
Into a Dutch oven pot on medium heat, add butter. When butter is melted, add onions. Saute 3 minutes, or until lightly browned.
To the pot with onions, add cashews, potatoes, broccoli, and chicken-style broth.
Bring just to a simmer; do not boil.
Cover and cook 10 minutes, or until potatoes and broccoli are tender. Remove from heat.
Stir in pesto.
Transfer soup to a high-speed blender. Blend the soup until smooth or to desired consistency. Depending on the size of your blender, you may need to do this step in two batches.
Add parmesan cheese and stir gently to blend.
Season with salt and pepper, to taste.
To serve: Ladle into soup bowls and top with additional parmesan.