Bring a large pot of water to a rolling boil. Add 1 tablespoon kosher salt, then add the pasta. Cook 1 minute less than the package directions for al dente. Reserve 1/2 cup pasta water before draining. Drain and set aside.

While the pasta cooks, pat the chicken breasts dry with paper towels. Season both sides with 1 teaspoon salt and 1/2 teaspoon black pepper.

Heat olive oil in a large skillet over medium-high heat until shimmering. Add the chicken breasts in a single layer. Cook without moving for 5-6 minutes, until the bottom is deep golden brown. Flip and cook another 5-6 minutes, until the internal temperature reaches 165°F at the thickest part. Transfer chicken to a cutting board and let rest for 5 minutes. Slice into bite-sized strips or cubes.

Reduce the skillet heat to medium. Add butter. Once melted, add garlic and cook, stirring constantly, for 30 seconds until fragrant. Do not let it brown.

Sprinkle the flour over the garlic and butter. Whisk constantly for 1 minute to cook out the raw flour taste. The mixture will form a thick paste (a roux).

Slowly pour in the chicken broth while whisking vigorously to prevent lumps. Bring to a simmer; the sauce will thicken slightly, about 2 minutes.

Reduce heat to low. Stir in the heavy cream, lemon zest, lemon juice, and lemon pepper seasoning. Whisk until smooth. Simmer gently for 1 minute — do not let it boil or the cream may separate.

Add the drained pasta, sliced chicken, and Parmesan cheese to the skillet. Toss to coat evenly. If the sauce is too thick, add reserved pasta water 1 tablespoon at a time until it reaches a silky consistency that coats the pasta.

Taste and adjust seasoning with salt or more lemon juice if needed. Serve immediately, garnished with parsley if using.