Prepare the chicken: If needed, slice the chicken breasts in half horizontally to make thin cutlets. Pat the chicken dry with paper towels. Set the heavy cream on the counter so it can come to room temperature while you prepare the rest of the recipe.
Season the chicken: Add the chicken to a medium bowl and toss with 1 tablespoon olive oil and 1 to 2 tablespoons blackened seasoning until evenly coated. Let the chicken rest at room temperature for 15 to 30 minutes, if time allows.
Boil the pasta: Bring a large pot of salted water to a boil. Add the fettuccine and cook until al dente according to package directions. Before draining, reserve 2 to 3 cups of the starchy pasta water. Drain and set the pasta aside.
Cook the chicken: Heat the remaining 1 tablespoon of olive oil and 2 tablespoons of butter in a large skillet set over high heat. Once the pan is HOT, add the chicken and cook for 4 to 6 minutes per side, or until deeply browned and fully cooked. If the chicken starts to burn, reduce the temperature as needed. Use a digital meat thermometer to check the internal temperature of the chicken. It should be removed from the pan and set aside when it registers 160-165℉. Cook the shallots and garlic: Return the same skillet to medium heat. Melt the remaining 4 tablespoons butter, then add the minced shallots. Cook for 4 to 5 minutes, stirring often, until softened. Add the garlic and cook for 30 seconds, just until fragrant.
Deglaze the skillet: Pour 1 cup of the reserved pasta water into the skillet and use a wooden spoon or spatula to scrape up any browned bits from the bottom of the pan. Bring to a gentle simmer, then reduce the heat to low.
Make the Alfredo sauce: Slowly stir in the room temperature heavy cream, then immediately add the cooked pasta. Toss continuously until the sauce begins to come together and lightly coat the noodles. Add more reserved pasta water as needed to loosen the sauce and help it emulsify.
Add the Parmesan cheese: Remove the skillet from the heat and stir in the Parmesan cheese until fully melted, smooth, and creamy.
Slice the chicken and serve: Slice the rested chicken into thin strips and serve over the Alfredo pasta. Season with additional salt and black pepper, if desired, and garnish with fresh basil before serving.