Add one tablespoon of the oil to a baking tray. Lay the asparagus on the tray, then coat them thoroughly with the oil. Season with a little salt. Cook in the middle of the oven until the asparagus change colour, about 8-10 min, depending on their size. Shake the tray a couple times during cooking, so that the asparagus will grill evenly. Take the asparagus out of the tray then cut them into 5 cm pieces.
Meanwhile, heat the remaining oil in a pan over medium-heat. Add the shallot, then sauté, with stirring, until it becomes translucent, 2-3 min. Add the mushrooms, then sauté until the mushrooms are browned and any liquid they give off has evaporated, about 4 min. Add a little salt, then set aside.
Cut the bread into cubes and dice the ham into small pieces. Randomly place the bread, ham, and vegetables into the prepared baking dish. Set aside.
Beat the eggs in a large bowl. Pour in the milk, mustard, and half of the cheese. Season with a little salt and pepper. Mix well, then pour the mixture into the prepared dish and over the bread and vegetables. Cover with the remaining cheese. Let stand at least 3 h in the refrigerator, up to overnight.
Take the baking dish out of the refrigerator at least 1 h before baking. Preheat the oven to 175ºC/350ºF.
Cover the dish with aluminum foil then put it in the middle of the oven. Bake 20 min. Uncover and bake an additional 25 min, until the omelet is firm and the top has formed a nice golden crust. Check with a toothpick or knife to see if it is cooked through. If it comes out clean, the omelet is ready.
Let the omelet rest a few minutes before cutting it into serving size pieces. Serve.
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