Heat the oven to 375F. Trim the green beans and cut them in half. Bring a large pot of salted water to a boil, add the beans, and cook 4 minutes. Drain and rinse under cold water to stop the cooking. Set aside.

Slice the shallots into thin rings and toss them with the all purpose flour. Heat the olive oil in a large skillet over medium heat. Fry the shallots in two batches, stirring often, 5 to 7 minutes per batch, until golden and crisp. Move them to a paper towel and sprinkle with a pinch of salt. Do not walk away; they go from golden to burnt in under a minute.

Wipe out the skillet. Melt the butter over medium-high heat. Slice the mushrooms and add them with a pinch of salt. Cook 8 to 10 minutes, stirring now and then, until they release their liquid and it cooks off and they brown.

Add the minced garlic and thyme and cook 1 minute. Sprinkle the whole wheat flour over the mushrooms and stir for 1 minute so the raw flour taste cooks out.

Pour in the broth a little at a time, stirring so no lumps form, then the milk. Simmer 3 to 4 minutes, stirring, until the sauce coats a spoon. Turn off the heat.

Stir in the Parmesan, then the Greek yogurt, salt, and pepper. Taste; it should be well seasoned because the beans dilute it.

Fold the green beans into the sauce and spread everything in a 9x13 baking dish. Bake uncovered 20 minutes, until bubbling at the edges.

Scatter the crispy shallots over the top and bake 5 minutes more. Let it rest 5 minutes before serving.