Clean the mushrooms and finely chop them in a food processor.

In a skillet, heat the olive oil and butter. Add the mushrooms and sauté until most of the liquid evaporates. Then add the onion and carrot (also chopped in the food processor). Sprinkle with a little salt to help release their juices. Cook over high heat, stirring often, until all the liquid is gone.

When done, set aside to cool slightly. For faster cooling, spread the mixture in a large bowl. Add the parsley, balsamic vinegar, breadcrumbs, and finally the egg (make sure the mixture has cooled before adding the egg). If the mixture is too loose, add more breadcrumbs; if it's too stiff, add another egg.

Mix by hand until everything is well combined and the mixture is firm, like ground meat. Shape into patties.

Preheat the oven to 400° F (200° C). In a skillet with a little olive oil over high heat, sear the patties for 1-2 minutes on each side until browned. Then transfer them to a baking sheet lined with parchment paper, or leave them in the skillet if it's oven-safe, and bake for 15 minutes. Top each patty with a slice of cheese and serve on buns with tomato, onion, and fries, or enjoy them as they are.

Enjoy!