1Sprinkle chicken thighs with 1/8 teaspoon pepper and wrap each with a slice of prosciutto.
2Heat 1 teaspoon oil in a medium 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.
3Reduce heat to medium and add the remaining 1 teaspoon oil, shallot and 1 1/2 teaspoons 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, 3 to 5 minutes. Add Marsala, return heat to medium-high and cook 2 minutes.
4Whisk broth, cornstarch and the remaining 1/8 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, 2 to 3 minutes more. Serve the chicken with the mushroom sauce, sprinkled with the remaining 1/2 teaspoon oregano.