Blanch: Heat the oven to 375°F (190°C). Boil the green beans in salted water for 4 minutes, shock in ice water, and pat completely dry.
Sauce: Brown the mushrooms in butter and oil, add shallot, garlic, and thyme, then stir in the flour. Add broth and milk and simmer until thick.
Finish sauce: Off the heat, cool 2 minutes, then stir in the yogurt, Parmesan, soy sauce, and seasonings.
Assemble: Fold the beans into the sauce, spread in a 2-quart dish, and top with the shallots tossed with panko, Parmesan, and oil.
Bake: Bake 20 to 25 minutes, until bubbling and golden. Rest 5 minutes.