Spray the bottom of a slow cooker with cooking spray.
Season chicken with salt, pepper, and poultry seasoning. Slice carrot into 1/2-inch thick slices.
Place chicken in the slow cooker. Add beans and carrots.
Pour the soup and broth over the chicken and veggies. Cook on high for 1 1/2 hours.
Once the hour and a half is complete, mix the dry ingredients for the dumplings together with the oil. Add the milk and stir until combined. Scoop batter by tablespoons into the slow cooker, allowing soup to coat the dough. Cover and cook an additional 2 1/2 hours.
Once done, carefully remove and shred the chicken; return back to the slow cooker and stir to combine.