Brown mushrooms. Heat oil in a small Dutch oven over medium-high heat. Once shimmering, add onion, garlic and mushrooms. Cook, stirring occasionally, until onions are just starting to brown and mushrooms are golden, about 10 minutes. Season with salt.
Crush tomatoes. While mushrooms are cooking, pour tomatoes into a large bowl and crush the whole tomatoes with your hands over the bowl.
Simmer sauce. Add tomatoes, thyme, anchovy paste, and 1/2 cup of water and bring to a boil. Cover, reduce heat to a low simmer and cook until slightly thickened, about 30 minutes. Remove thyme sprigs. Season with the 2 teaspoons of salt. Taste and add more if needed.
Cook pasta. While sauce is simmering, bring a large pot of heavily salted water to a boil. Once boiling, add pasta and cook according to package instructions.
Toss and serve. Remove pasta from cooking water with tongs and add into sauce. Toss to combine, if sauce seems too thick add a ladle or two of pasta water. Taste and add more salt if necessary. Serve with grated parmesan cheese and a sprinkle of fresh oregano over the top.