In a 10" cast iron skillet over medium heat, cook chicken sausage until just done (no longer pink). Add butter. When melted, add flour and stir 1-2 minutes to cook out the raw-flour taste. Slowly add milk and broth and stir; reduce heat and simmer until thick, 2-3 minutes.
Off heat, add salt, pepper, sage and shredded chicken. Stir well, cover and set aside.
In a large bowl, whisk flour with baking powder, baking soda, salt, thyme, sage and cayenne. Stir in cheese and butter. Add buttermilk and stir until it forms a dough, then knead 30-60 seconds until it just comes together. Roll into a 1/2" thick sheet. Cut with a 3" biscuit cutter; you will need at least 9 biscuits, so you may need to re-roll the scraps to get enough. Cut by pushing your cutter straight down, then twisting.
Place one biscuit in the center of the chicken mixture (still in the cast iron pan), then arrange the remaining biscuits around the outer edge of the pan. Place in preheated 400F oven for 20 minutes until biscuits are browned. DO NOT open the door. Cool 10 minutes before serving.