Preheat oven to 350ºF. Wash chicken and pat dry with a paper towel. Lightly coat in olive oil and season with salt and pepper. Place on a baking sheet and cook for 40 minutes.
Remove from oven and, once cool, shred chicken into bite-size pieces. Set aside. In a large stockpot, melt 1 tbsp butter and 1 tbsp olive oil on medium heat. Add diced carrots, celery, onion, and fennel. Stir to coat and cook for 3-5 minutes or until the vegetables begin to soften and the onion is translucent. Add the savory and thyme. Cook for 1 minute. Add dried dill, a generous pinch of sea salt, and some black pepper. Stir. Add chicken stock to the vegetables.
Bring to a boil and reduce heat to a simmer. Cook vegetables in the stock for 30 minutes or until tender. Add spinach, shredded chicken, and fresh dill. Stir to combine. Taste and season as necessary with salt and pepper. Place 1/2 C of brown rice in each individual soup bowl. Pour soup over brown rice and serve immediately.