Mix 1 1/3 cup flour, salt and pepper in food processor. Add cold shortening and butter. Pulse until mixture resembles coarse meal. with motor running, pour 3 T water through feed tube. Process 1 minute. Press pastry into 1/2 inch disk. Wrap in waxed paper. Chill at least 30 minutes.
Melt 3 T butter. Stir in 3 T flour and thyme. Cook, till golden. Whisk in 1 1/2 broth and all the cream until thickened. Stir in lemon juice, salt and pepper and remove from heat. Reserve.
Preheat oven to 350°F In large skillet, over med-high heat, melt remaining butter. Add onion; sauté 3-4 minutes. Add carrots, mushrooms and remaining 1/2 cup broth; cook stirring freq, 10-12 minutes until veggies are tender and broth has evaporated. Stir in peas, chicken, and reserved cream mixture; cook 2 minutes. Cover and keep hot over low heat.
Roll pastry between 2 sheets wax paper to 11 inch round. Transfer filling to 10-inch deep-dish pie plate. Peel off top sheet of paper from pastry; invert pastry over top of pie plate. Peel off bottom paper; crimp pastry. Brush with egg wash. Cut vents. Bake on baking sheet 30 minutes. Let stand 10 minutes before serving.