French Chicken and Mushrooms

I've been working night shifts in the ER this week.  This means eating (and sleeping) at strange times.  Do I eat breakfast when I wake up at 6pm?  Or is that dinner time?  Amidst this confusion I managed to throw this together with some chicken from the freezer and a package of  mushrooms in the fridge.

It may be the lack of sleep getting to me, but the use of Dijon mustard and thyme is enough to label a dish French, right?

French Chicken and Mushrooms (adapted from Food.com)
Feeds 3-4

1 package of mushrooms, washed and sliced
1 tablespoon butter
1 tablespoon olive oil
1 pound of boneless, skinless chicken breasts
1 teaspoon Dijon mustard
3 tablespoons balsamic vinegar
3 (or more) cloves of garlic, minced
1/4 teaspoon thyme
1/4 cup chicken broth

In a frying pan, melt butter over medium heat.
Add mushrooms and cook until golden brown.
Whisk together remaining ingredients to make sauce.
While mushrooms are frying, heat a skillet to medium and add olive oil.
Sear chicken on both sides.
Pour liquid in to skillet and cook until chicken is no longer pink.
Remove chicken.
Crank up the heat and reduce the liquid to a sauce.
Add chicken and mushrooms.
Stir to coat with sauce and serve.