In boiling salted water, simmer chicken tenderlions, broccoli, carrots, celery, and leeks until chicken is fully cooked and vegetables are tender, but still firm (about 15 minutes). Drain well and set aside.
In a medium sized skillet, melt butter and add flour. Cook for 5 minutes and add milk or half-and-half. Bring to a boil and simmer until thickened (about 10 minutes). Dissolve chicken boulillon cubes in 1/4 cup warm water and add to sauce. Blend in grated Parmesean cheese and remove from heat. Add cooked chicken and vegetables and mix well. Pour mixture into a deep-dish pie plate and cover with pie dough. Seal edges well and brush top with beaten egg. Bake until crust is golden brown and the filling is hot (about 45 to 50 minutes). Brushing a little milk on the top of the pie dough before baking will give the crust a delightful color.