Hello Dave! Running out of eggs right when you are ready to bake can be frustrating, but fortunately you have several great household alternatives that will give your savory pies a lovely finish.
For savory pies, heavy cream or whole milk is one of the absolute best choices. Brushing heavy cream or milk onto the top crust encourages beautiful browning because the milk proteins and lactose caramelize under the heat of the oven. Heavy cream in particular yields a richer golden brown finish and a soft sheen due to its higher fat content.
Melted butter or olive oil is another fantastic option. Brushing melted butter over the top pastry creates a crisp, golden crust with a subtle gloss and a rich flavor that pairs wonderfully with savory fillings. Olive oil works similarly and provides a nice light sheen.
If you happen to have a can of chickpeas in your pantry, you can use aquafaba, which is the liquid from the can. Brushing a light layer of aquafaba onto your pie crust produces an impressive glossy finish that comes remarkably close to a traditional egg white wash without altering the taste of your savory filling.
Lastly, you can thin out a small spoon of plain yogurt or sour cream with a drop of water or milk to make an easy wash. Whichever option you choose, apply it lightly with a pastry brush just before baking for the best results.