Bring a large pot of salted water to a boil. Cook linguine until al dente according to package directions. Reserve 1 cup of pasta water, then drain.
While pasta cooks, warm olive oil in a large skillet over medium heat. Add minced garlic and red pepper flakes. Cook just until fragrant, about 30 seconds.
Reduce heat to low. Pour in heavy cream, lemon zest, and lemon juice. Stir to combine, then let it gently warm for 1 minute. Do not boil.
Gradually sprinkle in Parmesan, whisking constantly until smooth and melted. Season with salt and a generous crack of black pepper.
Toss in the spinach and sun-dried tomatoes. Stir gently until spinach wilts, about 1-2 minutes.
Add the drained linguine to the skillet. Toss well with tongs until every strand is coated. If sauce seems too thick, add reserved pasta water a tablespoon at a time until silky.
Divide among bowls. Top with extra lemon zest, a dusting of Parmesan, and a drizzle of olive oil if desired. Serve immediately.