Melt butter and heat oil in a pot over high heat. Add onion and garlic, cook for 1 minute.
Add chicken and cook until it changes from pink to white.
Turn stove DOWN to medium. Add flour and stir for 1 minute.
Add half the milk, stirring as you go, and once incorporated into the flour add the rest of the milk and stir.
Add broth, rice, thyme and a just pinch of salt and pepper.
Stir, bring to simmer, then cover. Turn down to medium low and cook for 12 minutes.
Remove lid – there should still be liquid on the surface. Working quickly, add broccoli, push down so it’s partly submerged, cover and cook for a further 3 minutes until broccoli is just cooked.
Preheat grill/broiler. Remove lid and stir through half the cheese. Smooth surface, top with remaining cheese.
Grill/broil until melted (or just put lid on and allow to melt with residual heat on turned off stove).
Serve immediately, garnished with parsley if desired.
