Bring a large pot of generously salted water to a boil. Cook linguine according to package directions until al dente. Reserve 1 cup of pasta water, then drain and set aside.
While pasta cooks, season chicken cutlets with salt, pepper, and Italian seasoning. Heat olive oil in a large skillet over medium-high heat. Sear chicken for 4-5 minutes per side until golden and cooked through (165°F internal). Transfer to a plate and cover loosely with foil.
In the same skillet, reduce heat to medium. Melt 1 tablespoon butter and sauté garlic for 30 seconds until fragrant. Pour in wine (or broth), scraping up any browned bits. Simmer for 2 minutes until reduced by half. Add heavy cream, lemon zest, lemon juice, and red pepper flakes. Whisk and let it gently bubble for 3 minutes.
Reduce heat to low. Stir in remaining 2 tablespoons butter and Parmesan until melted and smooth. Add drained pasta and toss well to coat. If sauce is too thick, add reserved pasta water a little at a time until you reach a velvety consistency.
Slice rested chicken against the grain. Divide pasta among bowls, top with sliced chicken, drizzle with any extra sauce from the pan, and garnish with parsley and extra lemon zest. Serve immediately with more Parmesan.