Bring a large pot of salted water to a boil. Add the linguine and cook according to package directions until al dente. Just before draining, reserve 1/2 cup of the starchy pasta water.
While the pasta cooks, pat the chicken cutlets dry with a paper towel. Season both sides with garlic powder, lemon zest, salt, and pepper.
In a large skillet, heat the olive oil and 1 tablespoon of butter over medium-high heat. Add the chicken in a single layer. Cook for 3-4 minutes per side, until golden brown and cooked through. The internal temperature should reach 165°F (74°C).
Squeeze the lemon juice over the cooked chicken in the pan. Transfer the chicken to a plate or cutting board and let it rest.
Reduce the heat to medium. Add the remaining 3 tablespoons of butter to the same skillet. Once melted, add the minced garlic and cook for about 30 seconds until fragrant, stirring constantly.
Add the drained linguine, the reserved 1/2 cup of pasta water, and the 3/4 cup of grated Parmesan to the skillet.
Toss everything together continuously with tongs until the cheese melts and the liquid combines with the butter to form a light, creamy sauce that coats the pasta.
If you want a richer sauce, stir in the heavy cream. Taste and adjust seasoning with more salt and pepper if needed.
Slice the rested chicken. Serve the pasta in bowls, topped with the sliced chicken. Garnish with fresh parsley and extra Parmesan cheese.