Pat the chicken dry and season both sides with 1/4 teaspoon of salt. Set up a breading station with three shallow dishes. Place the flour in the first, the beaten egg in the second, and the panko breadcrumbs mixed with Parmesan cheese in the third.
Dredge one piece of chicken in the flour, shaking off any excess. Dip it into the egg, letting the excess drip off. Finally, press the chicken into the panko mixture, coating both sides evenly.
Heat the olive oil in a large skillet over medium-high heat. Carefully place the breaded chicken in the skillet. Cook for 3 to 4 minutes per side, until golden brown and cooked through. The internal temperature should reach 165°F (74°C). Remove the chicken to a plate and keep warm.
Reduce the heat to medium. Add the minced garlic to the skillet and cook for 30 seconds until fragrant.
Pour in the white wine to deglaze the pan, scraping up any browned bits from the bottom with a wooden spoon. Let it bubble for about 1 minute.
Stir in the chicken stock, heavy cream, lemon juice, lemon zest, and the remaining 1/4 teaspoon of salt. Bring to a simmer and cook for 3 to 5 minutes, until the sauce has slightly thickened.
While the sauce simmers, cook the angel hair pasta in a large pot of salted boiling water according to package directions. Drain well.
Add the cooked pasta directly to the skillet with the sauce. Toss to combine everything. Serve the pasta immediately, topped with the crispy chicken (either whole or sliced). Garnish with fresh parsley and extra Parmesan cheese if you like.