Butter a 10-inch casserole dish using about a teaspoon of butter. Move the oven rack to the middle position and preheat the oven to 500 degrees F (260 C).
Bring a large pot of salted water to a rolling boil and the whole green beans.
Blanch the beans by bringing the water back up to a full. Drain the beans and rinse them under cold water to chill them enough to handle.
Trim the stem ends off of the beans and chop them up into 1.5-inch lengths.
Melt 1 tablespoon of butter in a frying pan over medium heat and sauté the onions, button mushrooms, shimeji mushrooms, and salt until the mushrooms are starting to caramelize (about 6-7 minutes).
Transfer the mushrooms to a bowl and add the remaining tablespoon of butter to the pan along with the flour. Cook the roux until there are no lumps of flour, but don't let it brown.
Turn down the heat and pour the milk in at once. Stir the mixture vigorously until there are no lumps of roux remaining.
Turn up the heat to medium and bring the béchamel to a boil to thicken. Add the sauteed mushrooms as well as the green beans to the pan, and stir to distribute evenly.
To assemble the Doria, spread an even layer of rice along the bottom of the buttered casserole dish.
Top with the mushroom and green bean béchamel.
Cover with an even layer of shredded cheese.
Put the Doria in the oven and bake until the sauce is bubbly and the cheese has browned on top, about 15-20 minutes.