Coarsly chop the mushrooms.
Then process the mushrooms in a food processor, and reserve.
Without cleaning processor bowl, add the bread, milk, 1/2 teaspoon salt, and black pepper to taste.
Process until a paste is created.
Add the beef and process until all is well combined. About 6 pulses.
Finely dice the onion.
Heat the oil in a large saucepan over medium high heat. Add onion and mushrooms. Cook till veggies start to brown, and leave a fond on the bottom of the pan.
Add garlic and tomato paste. Cook till tomato paste starts to brown.
Add the liquid from the diced tomatoes and the oregano (reserve some of the oregano). Use the liquid to deglaze the pan, scraping up all brown bits.
Stir in the meat mixture, breaking up any chunks of meat. Cook until meat loses its color, but don't brown the meat.
Add the diced and crushed tomatoes.
Reduce heat to low, and simmer.
After about 30 minutes, when sauce thickens, stir in the cheese and reserved oregano. Adjust seasonings and serve.