For the topping:

Process the bread, butter, salt, and pepper in a food processor until coarsely ground. Transfer to a large bowl with the fried onions and set aside.

For the beans:

Heat the oven to 375°F. Bring 4 quarts of water to a boil in a large pot. Add 2 tablespoons salt and the green beans. Cook until bright green and crisp-tender, 4 to 5 minutes. Drain and plunge the beans immediately into a large bowl filled with ice water to stop the cooking process. Spread beans on towel-lined baking sheets to drain.

Meanwhile, cover the dried porcini with 1/2 cup hot tap water in a small microwave-safe bowl; cover with plastic wrap, cut several steam vents with a paring knife, and microwave on high for about 30 seconds. Let stand until the mushrooms soften, about 5 minutes. Lift the mushrooms from the liquid and mince; you should have about 2 tablespoons. Strain the soaking liquid through a fine mesh strainer (or coffee filter) and reserve.

Melt the butter in a large nonstick skillet over medium-high heat until the foaming subsides, about 2 minutes. Add the onion, garlic, button mushrooms, and cremini mushrooms and cook until the mushrooms release their moisture, about 2 minutes. Add the porcini mushrooms along with the reserved soaking liquid, thyme, 1 teaspoon salt, and the pepper and cook until the mushrooms are tender and the liquid has reduced to 2 tablespoons, about 5 minutes. Add the flour and cook for about 1 minute. Stir in the broth and reduce the heat to medium. Stir in the cream and simmer gently until the sauce has the consistency of dense soup, about 15 minutes.

To assemble and bake:

Spread the beans in a 3-quart gratin dish. Pour the mushroom mixture over the beans and mix to coat evenly. Sprinkle with the bread-crumb mixture and bake until the top is golden brown and the sauce is bubbling around the edges, about 15 minutes.

Serve immediately.