Bring a large pot of water to a rolling boil. Add 1 tablespoon salt and the pasta; cook according to package directions until al dente. Reserve 3/4 cup of the starchy pasta water, drain the pasta, and set aside.
While the water heats, pat chicken dry and season both sides with half the kosher salt and half the black pepper. Heat a large skillet over medium-high and add 1 tablespoon of olive oil. Sear the chicken 4–5 minutes per side until golden and cooked through (internal temp 165°F). Transfer to a plate and let rest 5 minutes, then slice into strips.
Reduce heat to medium and add the remaining 1 tablespoon olive oil to the skillet. Add the minced garlic and chopped sun-dried tomatoes and cook, stirring, for 1–2 minutes until fragrant — do not let the garlic brown.
Pour in the chicken broth to deglaze the pan, scraping up any browned bits. Let the broth reduce by about half, 2–3 minutes, then stir in the heavy cream and bring to a gentle simmer.
Lower the heat to medium-low and whisk in the Parmesan until smooth. Taste and adjust seasoning with remaining salt and pepper. If the sauce seems too thick, stir in a few tablespoons of reserved pasta water until you reach a silky consistency.
Return the sliced chicken to the skillet to warm for 1–2 minutes, then fold in the cooked pasta so it’s fully coated. Toss off the heat for 30–60 seconds to avoid curdling the cream and let the sauce thicken slightly as it cools.
Finish with the lemon zest, chopped basil, and red pepper flakes (if using). Serve immediately, passing extra Parmesan at the table.