Cut whites and dark green parts off leeks. Cut in half and then cut into small half-moon shapes. Place in colander and rinse well. Pat dry with paper towel before adding to pan. Chop garlic.
In large skillet, heat olive oil and two tablespoons of butter. Add leeks and chopped garlic to the skillet and saute for 3-5 minutes. Add one additional tablespoon of butter and flour. Stir flour into the butter, making sure all the flour has been stirred in.
Add wine to flour mixture, stirring.
Slowly add 32 ounces of chicken broth; whisk until thick with no lumps. When thickened, stir in half and half.
Add chicken, peas, and carrots; stir. Remove from heat.
Roll out pie dough; place in pie pan. Pour half of mixture into the crust. Roll out second crust and place on top. Roll the edges of top crust over the edges of the bottom crust and pinch edges together. Cut 4 slits on top of pie.
Repeat with second pie.
Wrap edges of the crust with foil. Bake at 325 for 45 minutes. Make an egg wash by whisking 1 egg and 1 tablespoon of water together. Remove pot pie from oven, take off the foil and brush pot pie with egg wash. Return to oven and bake for another 5-10 minutes until golden brown.