Pulse mushrooms in a food processor until finely chopped. Transfer to a small bowl. Wipe out food processor bowl.

Working in 3 batches, pulse cauliflower in food processor until pieces are about the size of a grain of rice (some smaller and some larger ones are fine), transferring to a medium bowl as you go.

Heat ¼ cup oil and 2 Tbsp. butter in a large heavy pot over medium-high. Add mushrooms and cook, stirring occasionally, until golden brown, 4–6 minutes. Add onion and 2 Tbsp. oil to pot. Cook, stirring occasionally, until onion is very soft and golden brown, 6–8 minutes. Add garlic, chile, and rosemary and cook, stirring occasionally, until garlic is softened and mixture is very fragrant, about 3 minutes. Add tomato paste and cook, stirring constantly, until paste is slightly darkened, about 2 minutes. Add cauliflower and cook, still stirring occasionally, until cauliflower is cooked down slightly and begins to stick to bottom of pot, 6–8 minutes. Season with salt, then keep warm over low heat.

Meanwhile, cook pasta in a large pot of boiling salted water, stirring occasionally, until almost al dente, about 1 minute less than package directions.s

Using a slotted spoon, transfer pasta to pot with sauce. Add 1 cup Parmesan, remaining 2 Tbsp. butter, and 1 cup pasta cooking liquid. Increase heat to medium and cook, stirring occasionally, until pasta is al dente and sauce is clinging to pasta, about 3 minutes. Remove from heat and stir in parsley. Taste and adjust seasoning with salt (it'll probably need another pinch or two). Finely zest lemon over pasta and toss once more.

Divide pasta among bowls. Top with more Parmesan, then drizzle with oil.