Step 1

Mix flour and salt in processor. Add butter and cheese; blend until moist clumps form. Shape dough into 6-inch-long log. Wrap; chill while making chicken and filling.

Step 2

Place all ingredients in large pot; bring to boil. Reduce heat; simmer until chicken is cooked through, about 45 minutes. Transfer chicken to large bowl; cool. Discard skin and bones. Cut meat into 1-inch pieces; return to bowl. Strain cooking liquid into large saucepan. Boil until reduced to 3 cups stock, about 1 hour.

Step 3

Heat 1 tablespoon oil in large skillet over medium heat. Add peppers and onion; cook until soft, about 6 minutes. Transfer to bowl with chicken. Heat 2 tablespoons oil in same skillet over medium-high heat. Add mushrooms; sauté until just brown, about 4 minutes. Add to bowl with chicken; mix in next 4 ingredients.

Step 4

Melt butter in large saucepan over medium heat. Add flour; whisk 2 minutes. Gradually whisk in 3 cups stock, then milk, cream and beef base; bring to boil. Season sauce with salt and pepper. Stir 3 cups sauce into chicken mixture (reserve remaining sauce for another use).

Step 5

Cut prepared dough crosswise into 6 equal pieces. Roll out pieces to 7-inch rounds on lightly floured surface. Divide filling among six 2-cup soufflé dishes. Cover each dish with 1 dough round. Press overhang to side of dish to adhere. DO AHEAD Can be made 1 day ahead. Cover; chill.

Step 6

Preheat oven to 350°F. Brush top of each dough round with egg glaze; cut 3 slits in each for steam to escape. Bake pies until crust is golden brown and filling is heated through, about 45 minutes.