Preheat your oven to 350 degrees Fahrenheit.
Optional: pound the pork chops thin: This step is completely optional, however, it will allow the pork chops to cook quicker. To do this simply place one pork chop in either a plastic bag or between a sheet of plastic wrap. Then pound the chop with a meat mallet until it is roughly a quarter of an inch thick. Repeat these steps with the remaining pork chops.
Salt & pepper all of the pork chops on both sides, then lightly dredge them in the flour.
Cook the pork chops: Get out a large oven-safe pan, pour the ¼ cup of olive oil in it, and place it over medium-high heat. Once the pan is hot place the pork chops in and let them fry for 2 to 3 minutes, or until they have become nicely browned. Then flip them over and fry for an additional 2 to 3 minutes. Once all pork chops are cooked, take them out of the pan and set them aside.
Cook the mushrooms & shallots: Add the minced garlic to the same pan and sautee them for a minute. Then toss in the shallots and sautee them for an additional 2 minutes. Finally, toss in the mushrooms along with the thyme and sautee them with everything for roughly 5 minutes.
Make the sauce: Add the wine and let it reduce for a minute. Then pour in the chicken stock and toss in the butter. Let this all cook and reduce for another 3 to 5 minutes.
Finish the dish in the oven: Take the pan off the heat and add the pork chops back in. Pour some of the sauce & mushrooms over them. Cover each pork chop with both kinds of cheese and place the pan in your oven. Let this bake until the cheese has melted to your liking.
Serve the pork lombardy along with the sauce and mushrooms, garnish with fresh basil and enjoy!