Bring a large pot of generously salted water to a rolling boil. Add the pasta and cook until 1 minute shy of al dente. Reserve 1 ½ cups of pasta water before draining.
Zest both lemons finely into a small bowl, then juice them into a separate bowl (you need about ¼ cup juice). Set the zest aside.
In a large skillet over medium heat, melt the butter. Add the garlic and cook until fragrant, about 30 seconds — do not let it brown.
Pour in the heavy cream and lemon juice. Whisk to combine. Let it simmer gently for 1-2 minutes until it thickens slightly and coats the back of a spoon.
Reduce heat to low. Add the grated Parmesan in small handfuls, whisking constantly, until fully melted and smooth. Do not let it boil or the sauce may become grainy.
Ladle in ½ cup of the reserved pasta water and whisk until the sauce is glossy and pourable. It should look thinner than desired — the pasta will absorb it.
Add the drained pasta directly to the skillet. Use tongs to toss the pasta in the sauce, adding more pasta water a splash at a time, until every strand is coated and the sauce moves freely (about 1-2 minutes).
Remove from heat. Sprinkle half the lemon zest over the pasta and toss once. Taste and adjust salt if needed. Divide into bowls. Top with remaining zest, a crack of black pepper, a pinch of red pepper flakes, and a few basil leaves placed deliberately on top. Serve immediately.