Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat. Add turkey and cook until lightly golden, 2 to 3 minutes per side. Transfer to a plate, cover and keep warm.
Heat the remaining 1 tablespoon oil in the pan. Add mushrooms and cook, stirring often, until browned, 4 to 6 minutes. Sprinkle with flour; stir to coat. Stir in broth and sherry; bring to a simmer. Continue simmering, stirring constantly, until the mixture is slightly reduced, 1 to 2 minutes. Add milk, peas and peppers; return to a simmer, stirring often. Cook until thick and slightly reduced, about 2 minutes. Stir in Parmesan and pepper. Return the turkey and any accumulated juices to the pan, turn to coat with sauce and cook until heated through, 1 to 2 minutes.
Ingredient note: Don't use the "cooking sherry" sold in many supermarkets — it can be surprisingly high in sodium. Instead, purchase dry sherry that's sold with other fortified wines in your wine or liquor store.