Prepare the chicken: Pat the chicken cutlets dry with paper towels. Season both sides with 1 1/2 tablespoons of the Cajun seasoning, 3/4 teaspoon of the salt, and all of the black pepper. Press the seasonings gently onto the surface.

Boil the pasta water: Bring a 6-quart pot of water to a rolling boil. Add 1 tablespoon of salt, then add the linguine. Cook according to the package directions until al dente, generally 9 to 11 minutes.

Reserve the pasta water: Before draining, scoop out 1/2 cup (120 ml) of the starchy cooking water. Drain the linguine thoroughly, but do not rinse it. Set it aside briefly while finishing the sauce.

Heat the skillet: While the pasta cooks, heat the olive oil and 1 tablespoon (14 g) of the butter in a 12-inch skillet over medium-high heat. The butter should foam without turning dark brown.

Cook the first chicken batch: Add half of the chicken cutlets in a single layer. Cook for 3 to 4 minutes, until the underside is browned and releases easily from the skillet.

Finish cooking the chicken: Turn the cutlets and cook for another 2 to 4 minutes. Transfer them to a clean plate when the thickest part reaches 165°F (74°C). Repeat with the remaining chicken, reducing the heat slightly if the seasoning begins to darken too quickly.

Rest the chicken: Let the cooked chicken rest for 5 minutes. Resting allows the juices to settle and keeps them from running out when the cutlets are sliced.

Sauté the garlic: Reduce the skillet heat to medium-low. Add the remaining 2 tablespoons (28 g) butter. Once melted, add the garlic and cook for about 30 seconds, stirring constantly. The garlic should smell fragrant but remain pale.

Deglaze the skillet: Pour in the chicken broth and scrape the skillet with a wooden spoon to release the browned seasoning. Add the remaining 1/2 tablespoon Cajun seasoning and remaining 1/4 teaspoon salt. Simmer for 2 to 3 minutes, until the liquid reduces slightly.

Add the cream: Pour in the heavy cream and stir until evenly combined. Bring the sauce to a gentle simmer over medium-low heat and cook for 3 to 4 minutes. It should lightly coat the back of a spoon but remain pourable.

Melt the Parmesan: Reduce the heat to low: Add the 1 1/2 cups Parmesan in three additions, whisking until each portion melts before adding the next. Do not allow the sauce to boil. It should look smooth, glossy, and free of visible cheese strands.

Toss with the linguine: Add the drained linguine and reserved 1/2 cup pasta water to the skillet. Toss gently with tongs for 1 to 2 minutes, until the sauce coats the noodles evenly. If the sauce looks loose, continue tossing over low heat for another minute. If it looks too thick, add a splash of warm water.

Finish with lemon: Remove the skillet from the heat and stir in the lemon juice. Taste the pasta before adding more salt because both the Cajun seasoning and Parmesan contribute seasoning.

Slice the chicken: Cut the rested chicken across the grain into strips. The interior should be opaque and moist, with no translucent areas.

Assemble and serve: Divide the creamy Parmesan linguine among six plates or shallow bowls. Arrange the sliced Cajun chicken over the pasta. Garnish with parsley and additional Parmesan, if using, and serve immediately.