Preheat the oven to 200 degrees F. Heat a large skillet over medium-high heat. Season the chicken with salt and pepper. Add 1 tablespoon oil to the skillet. Dredge 2 chicken cutlets in flour, shake off any excess and place in the skillet. Cook until golden, about 1 1/2 minutes per side; transfer to a baking dish. Repeat with the remaining 1 tablespoon oil and the other 2 chicken cutlets. Cover the dish loosely with foil; place in the oven while you prepare the vegetables.
Add the butter to the hot skillet, then add the scallions and mushrooms; cook, stirring occasionally, until the mushrooms brown, about 4 minutes. Pour in the broth and bring to a boil, scraping up any browned bits with a wooden spoon. Cook until the liquid is reduced by half, about 3 minutes. Add the cream and boil until the sauce thickens slightly, 3 to 4 more minutes. Stir in the snap peas and heat through; season with salt and pepper. Serve the chicken topped with the creamy vegetables.