Melt the butter in a large pot over low heat. Add the onion, salt and pepper and stir to combine. Put the lid on and let the onions sweat for 20 minutes. Add in the broth, using a wooden spoon to scrape up any brown bits of fond on the bottom of the pan. Add the chicken and broccoli and bring to a simmer. Cook until chicken is cooked through. 

Using a pair of tongs, remove the chicken and about half of the broccoli pieces and place in a bowl. 

Transfer the chicken to a cutting board and shred into bite-size pieces using 2 forks. Stir the cream into the pot. 

Toss the cheeses and arrowroot together, add to the pot and then blend using a hand immersion blender. Add the shredded chicken and remaining broccoli back to the pot and stir to combine. Season with sea salt and black pepper to taste. Serve.