Preheat oven to 300 degrees F.
In a large pan over medium high heat (I use my Le Creuset French oven), melt the butter and oil together. Add the cubed sirloin and brown, about 7-9 minutes. Remove meat to a covered baking dish and set aside, but do not wipe out the pan.
Add the onion, garlic, and sliced mushrooms to the pan the meat was browned in and stir well. Sauté until onion becomes slightly translucent, about 3-4 minutes.
Add the sautéed mixture over the meat (it's not necessary to stir them together with the meat) in the baking dish and cover. Bake in the preheated 300 degrees F oven for 90 minutes. Remove from oven and set aside.
In another large pot or pan (again, a big Le Creuset works well), combine the broth, vinegars, and soy sauce and bring to a simmer.
In a small bowl, combine the half and half, mustard, and cornstarch and whisk well. Gradually add the combination to the simmering mixture in the pot, stirring continuously, and simmer for 2-3 minutes until thickened nicely. (It's at this point that you should taste the sauce and decide whether or not you want more than 1 tablespoon of balsamic - I like it with both less and more balsamic -- it tastes different each way.).
Pour the contents of the baking dish into the pot, stir well, raise temperature to high, and cook for another 2 minutes while stirring until sauce is thickened. Remove from heat and season to taste with salt and freshly ground pepper.
Serve over hot cooked rice or egg noodles and enjoy!