Start by preheating your oven to 425°F so it's fully hot by the time your pies are ready to bake. While it heats, peel and dice your potatoes, slice your celery, and measure out the frozen vegetables. Having everything prepped ahead of time makes the cooking process move much faster.
Place the potatoes, celery, and frozen mixed vegetables into a large saucepan and cover with water. Bring it to a boil, then reduce to a simmer until the potatoes are fork-tender, about 10 minutes. Drain everything well in a colander and set it aside for later.
In the same saucepan, melt the butter and cook the onion and leeks until soft and fragrant, about 5 minutes. Whisk in the flour, salt, pepper, celery seed, onion powder, and Italian seasoning until it forms a smooth paste. This step is what gives the filling its rich, velvety texture.
Slowly whisk in the chicken broth and milk, pouring gradually so no lumps form. Keep whisking over medium heat until the mixture comes to a gentle simmer and thickens into a creamy gravy-like consistency. This usually takes about 5 to 7 minutes.
Remove the saucepan from the heat and stir in the cooked chicken along with the drained vegetables. Mix everything until it's evenly coated in that creamy sauce. At this point, you can use the filling right away or let it cool and freeze it for later.
Fit a rolled-out bottom crust into each pie plate, then divide the warm filling evenly between them. Top each pie with a second crust, pinch the edges together to seal, and cut a few small slits in the top to let steam escape. Brush lightly with cream or milk if you want a shiny, golden crust.
Place foil rings around the crust edges to prevent them from browning too fast, then bake for 30 to 35 minutes, or until the top is golden brown and the filling is bubbling through the slits. Remove the foil during the last few minutes if you want extra color on top.
Let the pies rest for about 10 minutes before slicing. This gives the creamy filling time to set slightly so it holds together when served rather than running all over the plate.