Bring the water to a boil in a large pot. Add the kosher salt, then add the fettuccine and cook until al dente, 9β11 minutes; reserve 1 cup of the starchy cooking liquid and drain the pasta.
Meanwhile, pat the boneless skinless chicken breasts dry and season all over with the black pepper. Heat the extra-virgin olive oil in a large skillet over medium-high until shimmering, 1β2 minutes, then sear the chicken until well browned and cooked through, 5β7 minutes per side (165Β°F internal). Transfer to a plate, tent loosely, and rest 5 minutes; slice across the grain.
Reduce the heat to medium, then add the unsalted butter to the same skillet. When melted and foaming, stir in the garlic and cook just until fragrant, 30β60 seconds; pour in the heavy cream and simmer, scraping up any browned bits, until slightly thickened and able to coat a spoon, 3β5 minutes.
Add the drained pasta to the skillet and toss to coat. Off the heat, gradually sprinkle in the Parmigiano Reggiano while tossing vigorously until the sauce is glossy and clings to the noodles; adjust with 0.25β0.5 cup of the reserved cooking liquid as needed.
Fold in the sliced chicken and any accumulated juices, then toss over low heat just to warm through, 1β2 minutes. Taste and adjust seasoning if needed.
Divide among warm bowls and finish with the flat-leaf parsley. Serve immediately.