Bring a large pot of salted water to a boil. Add the linguine pasta and cook according to package instructions until al dente, usually about 8-10 minutes. Drain the pasta, reserving about 1/2 cup of the pasta water, and set aside.
While the pasta cooks, heat 2 tablespoons of olive oil in a large skillet over medium-high heat. Season the sliced chicken breast with salt and pepper. Add the chicken to the skillet and sauté until golden brown and cooked through, about 5-7 minutes. Remove the chicken from the pan and set aside.
In the same skillet, lower the heat to medium and add the minced garlic. Sauté for about 1 minute until fragrant, being careful not to burn it.
Pour in 1 cup of chicken broth and 1 cup of heavy cream. Stir to combine and bring the mixture to a gentle simmer. Add the lemon zest and lemon juice, stirring well. Allow the sauce to reduce slightly for 3-4 minutes.
Stir in the grated Parmesan cheese until melted and the sauce thickens. Taste and adjust seasoning with salt and pepper as needed.
Return the cooked chicken to the skillet, then add the drained linguine. Toss everything together, adding reserved pasta water a little at a time if the sauce feels too thick. Heat through for another 1-2 minutes, ensuring the pasta is fully coated.
Remove from heat and sprinkle fresh parsley over the top. Serve immediately for the best flavor and texture.