In a Dutch oven, heat oil over medium heat. Add garlic and bacon, and cook, stirring occasionally, until bacon fat is rendered and garlic slightly browns, 10 to 12 minutes. Add peperoncino, letting it toast slightly, before stirring in.
Add cabbage and mushrooms; season with salt. Briefly stir vegetables, cover, reduce heat slightly, cook for 4 minutes, and stir again. Repeat until vegetables have wilted and are mixed together. Continue to simmer, covered, stirring occasionally, for about 1 hour, adding 1/2 cup broth or water at a time, as necessary.
Uncover pan, and cook, stirring frequently and moistening sauce with broth or water as needed, until cabbage is completely soft; remove from heat. Use immediately, or refrigerate in airtight containers for up to 1 week.
Bring a large pot of water to a boil; add salt. Add pasta; cook until al dente. Drain pasta, and transfer to cabbage mixture; toss to combine. Serve immediately sprinkled with cheese.