Cook angel hair pasta according to package instructions, drain, and set aside.
In a bowl, toss the hot cooked pasta with 2 tablespoons butter and the lemon zest until butter melts; stir in 1/4 cup grated Parmesan and season with salt and pepper to taste.
Preheat the oven to 375°F (190°C).
Place the beaten egg in one shallow bowl. In a second shallow bowl, combine 3/4 cup Panko, 1/4 cup grated Parmesan, 1 teaspoon dried basil, and salt and pepper.
Heat 1/4 cup (1/2 stick) butter in a large oven-safe skillet over medium heat; add 1/2 cup lemon juice and 1 teaspoon minced garlic and warm until fragrant but not boiling.
Dip each thin chicken breast first in the beaten egg, then press into the breadcrumb mixture to coat evenly.
Add the breaded chicken to the hot skillet and cook until golden brown, about 3–4 minutes per side depending on thickness; transfer chicken to a plate.
Spoon the lemony pasta into the skillet over the sauce, then arrange the cooked chicken breasts on top.
Top each chicken breast with optional lemon slices, sprinkle with 3/4 cup shredded mozzarella, and season with salt and pepper to taste.
Bake in the preheated oven until the chicken is cooked through and the cheese is melted, about 15 minutes.
Serve each chicken breast on a bed of the lemon angel hair pasta.