Add the olive oil in a medium frying pan. Lightly fry the potatoes, enough for the edges to get golden. Lay potato slices on a paper towel to absorb the extra oil. Repeat the same process for the eggplant slices. Set aside for later.
Prepare the bacon.
Cook the bacon slices unti crispy. Transfer to a paper towel to absorb the extra fat.
Make the bechamel sauce.
Melt the butter in a saucepan. Stir in the flour and cook, stirring constantly, until the paste cooks and bubbles a bit, but don't let it brown — about 2 minutes.
Add the warm milk, continuing to stir as the sauce thickens. Add salt and pepper to taste and a pinch or two of nutmeg, lower the heat, and cook, stirring for 2 to 3 minutes more.
Remove from the heat. To cool this sauce for later use, cover it with wax paper to prevent a skin from forming.
Roll the dough
Fill your galette
Leaving an inch and a half border add a layer of béchamel in a circular pattern. Add a layer of potatoes. Add a layer of eggplant. Lightly season with salt and pepper. Sprinkle some more béchamel sauce, a few pieces of bacon and parmesan cheese. Repeat this process 3 times. (Or as many as you like, all you need is extra potato and eggplant slices).
In a small bowl beat the egg with 1 tbsp of water to make the egg wash.
Fold the crust edges over the potatoes and eggplant. Lightly brush them with egg wash. Sprinkle fresh ground pepper all over, some more pieces of bacon and parmesan cheese.
Bake in a 350 F oven for 40 min or until crust is golden.
Serve warm with béchamel sauce on the side.
Notes
The béchamel ingredients make about a cup of sauce.