Slice each chicken breast horizontally into thin cutlets.
Sprinkle both sides of the chicken slices generously with salt and freshly cracked pepper.
Warm two tablespoons of olive oil in a large high-sided sauté pan or rondeau over high heat until the oil begins to shimmer and emits a light smoke.
Carefully lay the chicken pieces in the hot pan and sear for 4 to 5 minutes. Turn them over and continue cooking over medium heat for another 4 to 5 minutes, or until both sides are richly browned and the interior is cooked through. Remove the chicken from the pan and keep aside.
Into the same pan, pour in the remaining tablespoon of olive oil and stir in the minced garlic and diced shallot. Sauté over medium heat for 30 to 45 seconds until fragrant.
Lower the heat slightly and pour in the heavy cream. Let the cream warm through for 2 to 3 minutes without boiling. Then, gradually stir in the Parmigiano-Reggiano until it melts completely and the sauce thickens, about 3 to 4 minutes. Season the sauce with salt and black pepper, adjusting to taste. Set aside over low heat.
Meanwhile, in a separate large pot of vigorously boiling salted water, cook the fettuccine following the package guidelines until al dente, stirring occasionally to prevent sticking.
Once cooked, drain the pasta but reserve about a cup of the starchy water. Add the drained fettuccine to the pan with the Alfredo sauce, tossing gently to coat all strands evenly. If the sauce appears too thick, mix in a ladle or two of the reserved pasta water to loosen it to the desired consistency.
Slice the reserved chicken and serve it over the sauced fettuccine.