Preheat the oven broiler. Prepare a large nonstick skillet for cooking the chicken and an oven-safe skillet (cast iron or stainless steel) for assembling and broiling.
In a shallow bowl, combine 3 tablespoons of the grated Parmesan, breadcrumbs, oregano, basil, and 1/8 teaspoon of the salt. Place the beaten egg white in a separate shallow bowl.
Heat the butter and olive oil together in the nonstick skillet over medium heat until the butter is melted and the oil shimmers.
Dip each chicken tender into the egg white, then press into the breadcrumb mixture to coat evenly.
Add the coated chicken to the heated skillet and cook about 3 minutes per side, until the chicken is cooked through and golden brown; transfer to a plate.
Place the cooked, drained angel hair pasta into the oven-safe skillet and arrange the cooked chicken on top of the pasta.
In a medium bowl, stir together the marinara, balsamic vinegar, black pepper, and the remaining 1/8 teaspoon salt. Spoon the sauce evenly over the chicken and pasta.
Sprinkle the remaining grated Parmesan and the shredded Provolone or mozzarella evenly over the top.
Broil 2 to 4 minutes, watching closely, until the cheese is melted and bubbly. Remove from the oven and serve immediately.