Preheat oven to 325°F (165°C).
In a large Dutch oven, heat the oil over medium high heat.
Place the short ribs meaty sides down in the pan, working in batches if needed to avoid overcrowding.
Sear until they are nicely browned on each of the meaty sides (no need to brown the rib side).
Transfer the browned ribs to a plate and set aside.
In the same Dutch oven, reduce your heat to medium, and add in your onion, carrot, celery, thyme, and bay leaves.
Stir with a wooden spoon to scrape up all the browned bits in the pot, and continue cooking until vegetables are softened and translucent.
Add in the balsamic vinegar, marsala, and wine and stir, bringing heat up to high and letting the mixture come to a boil.
Let the liquid boil until it is reduced in half.
Pour in the stock and bring liquid back up to a boil.
Place the short ribs back in the pan placing the mushrooms on top and cover.
Remove ribs and mushrooms from the Dutch oven. Increase temperature of oven to 400°F (205°C).
Place ribs meat side up on a sheet pan and roast for 10 minutes or until browned.
While the ribs are roasting, strain the braising liquid through a fine sieve placed over a bowl to retain all the strained liquid. Press down on the solids in the sieve to squeeze all the liquid out. If you want you can skim off the fat at this point.