Bring a large pot of water to a rolling boil. Season generously with 1 tablespoon coarse sea salt. Add spaghetti and cook for 2 minutes less than the package instructions (very al dente), as the pasta will finish cooking directly in the sauce.

Just before draining, carefully scoop out and reserve at least 1 cup of cloudy, starchy pasta water. Drain pasta and set aside.

While pasta boils, heat olive oil and 2 tablespoons of butter in a large skillet over medium-low heat. Add minced garlic and half of the fresh lemon zest. Sauté gently for 1 minute until fragrant without browning the garlic.

Pour in the fresh lemon juice and heavy cream. Whisk gently and let it simmer on low for 2 minutes until slightly reduced and fragrant. Season with freshly cracked black pepper.

Transfer the hot drained spaghetti directly into the simmering skillet. Pour in 1/2 cup of reserved starchy pasta water and the remaining 2 tablespoons of cold butter. Using tongs, swirl and toss the pasta vigorously over medium heat for 2 minutes until a glossy, creamy sauce coats every single strand.

Remove skillet from the heat completely. Sprinkle in the grated Parmigiano-Reggiano cheese in three additions, tossing vigorously after each so the cheese melts smoothly into the emulsion without clumping. Add more pasta water a tablespoon at a time if sauce seems too thick.

Twirl pasta into shallow bowls, garnish with remaining lemon zest, fresh torn basil leaves, extra black pepper, and shaved parmesan. Serve immediately!