Preheat the oven to 400º F.

Season the chicken with salt and pepper. Heat the oil in a large nonstick skillet over medium-high heat. Add the chicken to the skillet; cook until golden, about 4 minutes per side. Transfer chicken to a shallow baking dish (do not clean skillet). Bake until the chicken is cooked through, about 15 minutes.

Meanwhile, heat the same skillet over medium-high heat. Add the onion, carrot, celery, and garlic and cook, stirring, until golden, about 5 minutes. Add the morels and continue to cook until softened, 5 minutes longer. Add the chicken broth and Madeira; boil until the gravy is reduced to 1 1/2 cups, about 5 minutes. Strain the gravy into a small saucepan; add the morels. Cover and simmer 5 minutes. Whisk 1 tablespoon cornstarch and remaining 1 tablespoon Madeira in a small bowl. Whisk into the gravy; simmer until thickened, about 30 seconds. Season gravy with salt and pepper. Spoon the mixture over the chicken and serve.