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 seasoning gently onto the surface.

Bring a 6-quart pot of water to a rolling boil. Add 1 tablespoon fine salt, then add the linguine.

Cook according to the package directions until al dente, generally 9 to 11 minutes. Before draining, reserve 1/2 cup (120 ml) of pasta cooking water. Drain without rinsing.

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.

Add half of the chicken cutlets in a single layer. Cook for 3 to 4 minutes, until deeply browned underneath and easy to release from the skillet.

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

Let the cooked chicken rest for 5 minutes while preparing the sauce.

Reduce the skillet heat to medium-low: Add the remaining 2 tablespoons (28 g) butter. Once melted, stir in the garlic and cook for about 30 seconds, until fragrant but not browned.

Pour in the chicken broth and scrape the bottom of the skillet to loosen the browned bits. Stir in the remaining 1/2 tablespoon Cajun seasoning and remaining 1/4 teaspoon salt.

Simmer for 2 to 3 minutes, until the liquid reduces slightly and smells deeply savory.

Pour in the heavy cream: 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.

Reduce the heat to low: Add the Parmesan in three additions, whisking until each portion melts before adding the next. Do not boil the sauce. It should look smooth and glossy.

Add the drained linguine and reserved 1/2 cup (120 ml) pasta water to the skillet. Toss with tongs for 1 to 2 minutes, until the noodles are evenly coated and the sauce clings to them.

Remove the skillet from the heat and stir in the lemon juice. Taste and adjust seasoning only if needed, since the Parmesan and Cajun seasoning contribute salt.

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

Divide the creamy Parmesan linguine among six plates: Arrange the sliced Cajun chicken over the pasta and garnish with parsley, extra Parmesan, and crushed red pepper flakes as desired.