For the Bechmel sauce, bring the oil and butter to high heat and stir in the flour a little at a time to avoid clumping.
Add the milk and, stirring continuously, the salt, pepper, shallots, garlic, and bay leaves.
Simmer for about 5 minutes, stirring occasionally. Discard the bay leaves and reerve.
For the lasagna, place the two flours and the salt in the bowl of a stand mixer. With the mixer on low speed, drizzle in the oil until a smooth dough ball is formed and pulls away from the side of the bowl. Add a little water OR a little flour if the dough is too dry OR too wet.
Turn the dough out onto a floured board and divide in to 8 or 10 manageable hunks.
Run each hunk through a pasta sheeter to create lasagna sheets approximately 5x9.
Pour enough of the Bechemel into a 5x9 baking dish to just cover the bottom and lay a sheet of pasta on top of it.
Cover this sheet with the chicken and cover that with another sheet of pasta.
Spread the mushrooms on top and cover with another sheet.
Continue the layering by adding the Manchego, pasta, Pecorino, pasta, Taco cheeses, and pasta.
Cover the layers with the remaining Bechemel sauce and top with the bread crumbs.