In a large Dutch oven over medium-high heat, heat 1 tbs oil. Brown the chicken on all sides, 5 to 7 minutes. Remove from pan and set aside.
Add the remaining oil to the pan and cook the onions and celery until soft, 5 to 7 minutes.
Sprinkle the flour, mustard, paprika, red pepper flakes, and pinch of salt and pepper over top and cook for 1 minute, stirring the whole time. Add in the thyme (either plucked or as a sprigs to pull out before serving).
Carefully, whisk in a bit of the milk. The flour will quickly absorb it. Whisk in the remaining milk slowly so that no lumps form. When smooth, whisk in the broth, in a slow steady stream.
Add the potatoes and bring to a boil. Reduce heat to a simmer and cook fork 10 minutes.
Add in the broccoli and the reserved chicken, along with any juices, and cook for another 10 minutes.
Lower heat to medium-low (if a simmer doesn't have you there already) and stir in the cheese. Stir until completely melted.
Serve with a garnish of thyme and more cheese if desired.