For the mushroom topping: Add the butter, oil, thyme, onions, chiles, salt and pepper to a pan over medium heat. Cook until the onions are tender and the thyme is fragrant, 6 to 8 minutes. Add the mushrooms and cook until they lose some moisture and wilt, about 5 minutes. Transfer the mushrooms to a plate and wipe out the pan to reuse.
For the patty: Whisk the Worcestershire, salt, pepper and garlic in a small bowl. Break up the ground chuck in a separate large bowl. Gently combine the Worcestershire and the ground chuck. Form the ground chuck into a ball.
Heat the pan over medium-high heat. Place the ball of ground chuck between two sheets of parchment or plastic wrap. Flatten the ball to a size a little larger than the flatbreads, using a rolling pin or the bottom of a glass. Place the flattened patty in the pan and cook on one side until the edges begin to curl under a bit. Flip the patty and cook the other side to your desired amount of doneness, about 5 minutes total.
Adjust an oven rack to the center of the oven and preheat to 350 degrees F. Place one of the flatbreads on a sheet of aluminum foil. Evenly layer the American cheese on top and sprinkle with the bacon. Place the patty on top and evenly spread the mushroom mixture on top. Place the smoked mozzarella on top. Place the second flatbread on top and tightly wrap the sandwich inside the foil and bake, flipping the sandwich halfway through the cooking process, about 10 minutes.
Remove the sandwich from the oven and let rest 5 minutes before opening. Cut the sandwich in to 4 wedges with a knife. Serve with ketchup.