Bring 4 quarts of salted water to a boil and cook fettuccine until al dente; reserve 1/4 cup pasta water before draining.
Pat steaks dry with paper towels and season both sides generously with salt and pepper.
Heat olive oil in a heavy cast-iron skillet over high heat until the oil reaches its smoke point.
Place steaks in the skillet and sear for 3-4 minutes per side until a deep Maillard crust forms.
Lower heat to medium, add 2 tablespoons of butter, 2 cloves of minced garlic, and rosemary sprigs to the skillet.
Tilt the pan and continuously spoon the foaming butter over the steaks for 2 minutes to reach an internal temperature of 135°F (57°C).
Transfer steaks to a cutting board and allow to rest for 8 minutes.
In a clean saucepan over medium heat, melt the remaining 2 tablespoons of butter and sauté the remaining garlic for 60 seconds.
Add heavy cream and simmer for 3-5 minutes until the volume is reduced by one-third.
Whisk in the Parmesan cheese until smooth; incorporate reserved pasta water if the sauce requires thinning.
Toss the cooked pasta into the cream sauce until evenly coated.
Slice the rested steak against the grain and plate immediately over the pasta.