Butterfly the chicken breast halves by cutting them horizontally. Pound to a uniform 1/4-inch thickness, then dredge in a mixture of flour, salt, pepper, and Parmesan until fully coated.

Heat olive oil in a skillet over medium-high heat. Add chicken cutlets and cook until golden brown on each side, about 3 minutes per side. Remove and keep warm under aluminum foil.

Melt 1 tablespoon of butter in a medium pan over medium heat. Add chopped onion and sauté until translucent, about 3 minutes. Stir in minced garlic and cook for 30 seconds until fragrant.

Pour in the Pinot Grigio and lemon juice, boiling the mixture over high heat. Cook until the liquid reduces to about 2 tablespoons, roughly 5 minutes.

Lower the heat to the lowest setting. Gradually whisk in the remaining butter, one piece at a time, ensuring each piece melts before adding the next. Season with salt and pepper, then fold in the rinsed capers.

Plate the cooked angel hair pasta as a base, lay the browned chicken on top, and spoon the lemon butter caper sauce over both.