In 12-inch skillet, heat 2 teaspoons of the oil over medium-high heat until hot. Add mushrooms; cook 2 to 3 minutes, stirring frequently, until tender. Remove mushrooms from skillet; place in bowl. Set aside.
In same skillet, melt butter and remaining 1 teaspoon oil over medium-high heat. Add chicken and carrots; sprinkle with salt and pepper. Cook 5 to 7 minutes, stirring occasionally, until browned. Stir in onions and wine. Heat to boiling. Reduce heat to medium-low. Cover; cook about 20 minutes or until chicken is no longer pink in center.
Meanwhile, heat oven to 450°F. Remove pie crust from pouch; unroll crust onto ungreased cookie sheet. Sprinkle evenly with cheese and thyme; roll in lightly with rolling pin. Prick crust generously with fork. With pastry wheel or sharp knife, cut into 12 wedges; separate slightly. Bake 7 to 10 minutes or until light golden brown.
In small bowl, mix flour and water until smooth. Add to juices in skillet; cook over medium heat, stirring constantly, until bubbly and thickened. Stir in whipping cream, peas and cooked mushrooms. Cook 3 to 4 minutes, stirring frequently, until thoroughly heated. Serve chicken and vegetables with pastry wedges.