Cut the tenderloin crosswise into 8 slices, each about 1 1/2 ounces. Pound the slices until they are about 1/4-inch thick, then sprinkle them with 3/4-teaspoon of the salt and pepper.
Beat the egg thoroughly with a fork in a shallow bowl.
Combine the bread crumbs and the cheese in another shallow bowl. Dip the slices of meat into the egg to lightly coat each side, let the excess drip off, and dip the meat into the crumb mixture until well coated on each side. Arrange the breaded meat on a platter.
Divide the olive oil between two nonstick skillets and heat until hot but not smoking. Arrange 4 scaloppine in each skillet so that they don't overlap and sauté over high heat for about 1 1/2 minutes on each side, or until nicely browned. Arrange on a clean platter, and keep warm in the oven.
Add the butter and onion to one of the skillets, and sauté onion over high heat for 1 minute or until slightly softened.
Add the mushrooms and the remaining 1/2-teaspoon salt and sauté for 2 to 3 minutes, until the mushrooms are lightly browned and most of their liquid has evaporated.
Spoon the mushrooms over the scaloppine, sprinkle with the chives and lemon juice, and serve.