Melt the butter in a pan over medium heat, add the onions and cook until a deep golden brown, about an hour, stirring occasionally, adding a bit of water or broth to deglaze the pan as needed.

Add the garlic, thyme and flour and cook for another minute while stirring.

Add half of the broth, deglaze the pan, bring to a boil, reduce the heat and simmer for 5 minutes.

Meanwhile, heat the oil in another pan over medium-high heat, season the chicken with salt and pepper, add the chicken and brown on both sides, about 2-4 minutes per side before setting aside.

Add the remaining cup of broth to the pan, deglaze it, mix in the balsamic vinegar and mustard, bring to a boil, reduce the heat and simmer to reduce by half, about 5 minutes.

Place the onions in an oven safe pan, top with the chicken and broth from the chicken pan, cover with a lid or foil and bake in a preheated 350F/180C oven for 20 minutes.

Sprinkle on the cheese and broil until it has melted, about 2-4 minutes.