Bring a large pot of well-salted water to a boil and cook the angel hair according to the package directions until it's al dente. Drain it in a colander, then toss it right away with a tablespoon of olive oil so the strands don't clump together. Set the pasta aside while you work on the chicken and sauce.
Heat the remaining tablespoon of olive oil in a large skillet over medium-high heat. Add the cubed chicken breast, season it with salt and pepper, and let it cook undisturbed for a couple of minutes so it browns properly. Once it's golden and cooked through, move it to a plate and set it aside.
Lower the heat to medium and melt the butter in the same skillet you used for the chicken. Add the minced garlic and stir it constantly for about a minute, just until it turns fragrant. Keep a close eye on it here, since garlic can go from golden to bitter in seconds.
Turn the heat down to medium-low and pour in the lemon juice and white wine, scraping up any browned bits from the bottom of the pan. Stir in the heavy cream and sliced basil, then season with a bit more salt and pepper. Let everything simmer gently for a couple of minutes until the sauce is heated through and slightly thickened.
Add the drained pasta straight into the skillet with the sauce and use tongs to toss it until every strand is coated. Return the seared chicken to the pan as well, folding it in gently so it warms back up alongside the noodles. Taste the sauce here and adjust the salt, pepper, or lemon juice if it needs it.
Plate the pasta while it's still warm and top it with freshly grated Parmesan and a scattering of fresh basil leaves. I like to add one more small squeeze of lemon right before serving for extra brightness. Serve it immediately so the sauce stays silky and doesn't stiffen up on the plate.