In a large pot, bring salted water to a boil. Add the 8 ounces of fettuccine and cook according to package instructions until al dente. Drain and set aside, reserving about 1 cup of pasta water for later use.
While the pasta is cooking, season the 1 pound of chicken breasts with salt and pepper on both sides. In a large skillet, heat 4 tablespoons of unsalted butter and 1 tablespoon of olive oil over medium-high heat. Once melted and hot, add the chicken breasts to the skillet. Cook for about 5-7 minutes on each side, or until the chicken is golden brown and cooked through. Remove from the skillet and let rest for a few minutes before slicing.
In the same skillet, lower the heat to medium and add 1 tablespoon of all-purpose flour. Whisk it together with the remaining butter and oil for about a minute to create a roux. Add 2 cloves of minced garlic and sauté for another minute until fragrant. Gradually pour in 1 cup of low sodium chicken broth, whisking to combine and prevent lumps.
Stir in 1 tablespoon of fresh thyme, 1 tablespoon of capers, and 3 tablespoons of freshly squeezed lemon juice. Let the mixture simmer for about 3-4 minutes, allowing it to thicken slightly.
Reduce the heat to low and slowly stir in 1 cup of heavy cream. Allow it to warm through for a couple of minutes, then season the sauce with additional salt and pepper to taste. If the sauce is too thick, add a splash of the reserved pasta water until you reach your desired consistency.
Slice the cooked chicken into strips and add it back to the skillet along with the drained fettuccine. Toss gently to coat the pasta and chicken in the creamy lemon sauce.
Plate the Creamy Lemon Chicken Piccata Fettuccine and garnish with chopped fresh parsley and a generous sprinkle of Parmesan cheese. Serve immediately while hot!