Sprinkle chicken thighs with 1/4 teaspoon pepper and wrap each with a slice of prosciutto.
Heat 2 teaspoons oil in a large nonstick skillet over medium-high heat. Add the chicken and cook, turning once, until browned on both sides and cooked through, 4 to 5 minutes per side. Transfer to a plate; tent with foil to keep warm.
Reduce heat to medium and add the remaining 2 teaspoons oil, shallot and 1 tablespoon oregano to the pan. Cook, stirring, until the shallot is beginning to brown, 1 to 2 minutes. Add mushrooms and cook, stirring, until browned in spots, 4 to 6 minutes. Add Marsala (or sherry), return heat to medium-high and cook 2 minutes.
Whisk broth, cornstarch and the remaining 1/4 teaspoon pepper in a measuring cup; add to the pan, stirring. Return to a simmer and cook, stirring constantly, until the sauce is thickened and glossy, about 4 minutes. Serve the chicken with the mushroom sauce, sprinkled with the remaining 1 teaspoon oregano.
Tip: Marsala, a fortified wine from Sicily, is a flavorful addition to many sauces. Don't use the “cooking Marsala” sold in many supermarkets-it can be surprisingly high in sodium. Instead, purchase Marsala that's sold with other fortified wines in your wine or liquor store. An opened bottle can be stored in a cool, dry place for months.