Bring a large pot of salted water to a boil. Cook the tagliatelle according to package directions. Drain and reserve.
While the pasta is cooking, heat a large sauté pan over medium-high heat. Add 1 TBSP of the oil and cook the onions until starting to brown and soften (about 5 minutes).
Add the cabbage and thyme and continue to cook, stirring often until beginning to brown in spots (about 3 minutes). Deglaze the pan with the apple cider. Cover and cook until wilted (2 minutes), then uncover and continue to cook until most of the cider has evaporated (another 5 minutes). Stir in the tagliatelle.
Heat a medium sauté pan over medium-high heat. Season pork with salt and pepper on both sides. Add the remaining 1 TBSP oil and cook pork until nicely browned on one side (about 2 minutes). Repeat on the second side. Remove to a plate and let rest while plating the pasta mixture.
Divide pasta between 4 plates. Top each with one piece of pork.