Heat 1 tablespoon oil in a large cast iron skillet over medium-high heat. When the skillet just starts to smoke, add the pork, two chops at a time, and cook until well browned, about 3-4 minutes on each side. Remove the pork chops to a rimmed baking sheet. Repeat with the remaining 2 pork chops.

While the pork chops are in the oven, make the pan sauce. Pour out the excess oil in the skillet. Add the butter and olive oil to same skillet used to brown the pork and heat on medium-high until the butter is melted. Add the mushrooms and a pinch of salt. Cook over medium-high heat, stirring occasionally, until the mushrooms release their moisture and begin to turn golden, 6 to 8 minutes. Add shallots; cook and stir for 2 minutes. Add the garlic; cook and stir for 30 seconds. Season with salt and pepper. Carefully add the white wine; cook and stir scraping up any browned bits stuck to the bottom of the skillet. Cook until most of the wine is evaporated, about 2 minutes.

Reduce the heat to medium and add the chicken broth, thyme leaves, and Dijon mustard. Transfer the chops to a plate and add the juices on the baking sheet to the skillet. Stir in the flour/water mixture and bring to a boil. Cook for about 3-5 minutes or until the sauce is thickened and slightly reduced. Remove from the heat and add the heavy cream if using. Salt and pepper to taste.