1. Process mushrooms in food processor until finely chopped, about eight 1-second pulses, scraping down side of bowl as needed; transfer to medium bowl. Add bread, milk, 1/2 teaspoon salt, and 1/2 teaspoon pepper to now-empty food processor and process until paste forms, about eight 1-second pulses. Add beef and pulse until mixture is well combined, about six 1-second pulses.

2. Heat oil in large saucepan over medium-high heat until just smoking. Add onion and mushrooms; cook, stirring frequently, until vegetables are browned and dark bits form on pan bottom, 6 to 12 minutes. Stir in garlic, pepper flakes, and tomato paste; cook until fragrant and tomato paste starts to brown, about 1 minute. Add 1/4 cup reserved tomato liquid and 2 teaspoons fresh oregano (if using dried, add full amount), scraping bottom of pan with wooden spoon to loosen browned bits. Add meat mixture and cook, breaking meat into small pieces with wooden spoon, until beef loses its raw color, 2 to 4 minutes, making sure that meat does not brown.

3. Stir in crushed and drained diced tomatoes and bring to simmer; reduce heat to low and gently simmer until sauce has thickened and flavors have blended, about 30 minutes. Stir in cheese and remaining teaspoon fresh oregano; season with salt and pepper to taste.

Makes about 6 cups